Universal Polynomials in Lambda rings and the K-theory
of the infinite loop space tmf
by
John R. Hopkinson
MMath. Mathematics
University of Warwick, 2001
Submitted to Department of Mathematics in partial fulfillmentof the
requirements for the degree of
Doctor of Philosophy
at the
Massachusetts Institute of Technology
June 2006
@2006 John R. Hopkinson. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute
publicly paper and electronic copies of this thesis document in whole or in
part in any medium now known or hereafter created.
Signature of Author
-J \
RL
Department of Mathematics
/
(/
May 1, 2006
//
/
Certified by
(
/
Pr
Michael J. Hopkins
/Vt
or of Mathematics, Harvard University
A
_
r
Accepted by
i8
Thesis Supervisor
,,,~~
avel I. Etingof
Professor of Mathematics
Chairman, Department Committee on Graduate Students
ARCHIVES
. ,, .
MASUCHUSE!TS
iSUE
OF TECHNOLOGY
1
J UN 0 5 2006
52LIBRARIES
LIBRARIES
Universal Polynomials in Lambda rings and the K-theory
of the infinite loop space tmf
by
John R. Hopkinson
Submitted to Department of Mathematics on 4th May, 2006 in partial
fulfillment of the requirements for the degree of Doctor of Philosophy
ABSTRACT
The algebraic structure of the K-theory of a topological space is described
by the more general notion of a lambda ring. We show how computations
in a lambda ring are facilitated by the use of Adams operations, which are
ring homomorphisms, and apply this principle to understand the algebraic
structure.
In a torsion free ring the Adams operations completelydetermine the lambda
ring. This principle can be used to determine the K-theory of an infinite loop
space functorially in terms of the K-theory of the corresponding spectrum. In
particular we obtain a description of the K-theory of the infinite loop space tmf
in terms of Katz's ring of divided congruences of modular forms. At primes
greater than 3 we can also relate this to a Hecke algebra.
Thesis Supervisor: Michael Hopkins
Title: Professor of Mathematics, Harvard University
." ,
,4!
2
Acknowledgements
When I thought about the people who I wanted to give my thanks to in these
acknowledgements, they fell into two groups: those who made it possible to
write this thesis and those who made it enjoyable. In no real order, they were
all the following. My advisor Mike, for his many suggestions, but mostly his
wit and wisdom. Mark and Haynes, for agreeing to sit on my committee, and
also for many corrections and stylistic suggestions, and advice through my
work at MIT. The Kennedy Memorial Trust and the department of maths for
all their support. Josh, for various productive conversations. The UMO staff,
who were always welcoming. Victor and Sigurdur, for making my qualifiers
a pleasure, and for always being around to sign my registration forms. Harriet,
who provided me with family in the US. My friends, who were willing to talk
any time I was recharging my brain, but are too numerous to list. Finally, I
want to thank the Bubblesss, for her helpful suggestions on the content; she'll
appreciate that I put them here.
3
Contents
1
Introduction
5
2
A-rings
7
2.1 The polynomials Pm,nand P
. . .
2.2
Adams operations ..........
2.3
The operation
2.4
The universal polynomials revisited
2.5
K-Theory
P
..........
examples
. . . . . . . . .
...............
...............
...............
...............
...............
3 Bousfield's Theorem
8
10
12
15
17
20
3.1
Z/2-graded p-adic A-rings and Z/2-graded p-adic p-modules.
21
3.2
The free OP-ringfunctors ....................
23
3.3
K*(QoE; Zp) .........................
25
3.4
Examples
28
............................
4 The K-homology of tmf
4.1
K-cohomology and K-homology .
4.2
The ring of divided congruences .
4.3
Formulae for the generators of D
4.4
D as Kotmf ...........
4.5
Diamond operators ........
5 The K-cohomology of tmf and QO°tmf
5.1
Hecke Algebras and duality
.....
5.2 K*(Q°tmf; Zp) for p> 2 .......
5.3 K*(Qf tmf; Z2) ............
..............
..............
..............
..............
..............
..............
..............
..............
32
32
33
36
41
45
46
46
50
52
A Combinatorial proof of the formula for P3,,
55
B The canonical differential
62
C Tables of Pn and Pm,n
67
4
1
Introduction
The main goal of this thesis is to describe the K-cohomology of the infinite
loop space Qftmf with p-adic coefficients.The algebraic structure of the Ktheory of a space is what is called a A-ring. This structure can be concretely
described if we consider the example of the K-theory of a finite CW-complex
X. In this case the group K°X is obtained by considering the monoid of vector
bundles on X under the operation of direct sum, and introducing additive
inverses. Specifically we take the free abelian group generated by the monoid
and quotient by terms of the form E + F-
E
F. A basic theorem, the
splitting principle, says that any element of K°X can be represented by (the
pullback of) a sum of line bundles. There are various operations on K°X
corresponding to the various constructions that can be performed on vector
bundles. For example it becomes a ring under the operation of tensor product.
There is also the operation of taking the n th exterior power. On a sum of line
bundles E =
1 Ei we have
An E = E
E
E(n)
n!oESm
rcsm
which, upon identifying tensor products in different orders, is the nth elementary symmetric function in the Ei (provided n < ra).
The operations An are not ring homomorphisms (at least for n > 1). The additive behaviour is relatively simple to describe but the multiplicative behaviour
is more complicated. However the theory of symmetric functions allows us to
describe the action of An on a product. Specifically, it is a classical theorem
that any symmetric polynomial can be written as a polynomial in the elemen-
tary symmetricfunctions.If we considera productE 0 F = (Z E/) 0 (Z Fj)
then the nth exterior power will be symmetric under permutations of the Ei
and under permutations of the Fj. It followsthat An(E 0 F) will be a polynomial in the Ak(E) and At(F) for k, < n. This polynomial is independent
of the number of summands, so long as there at least k, and is denoted Pn.
Similarly we can consider what happens if we take Am (AnE). The result will be
the mth elementary symmetric function in the sumnmands of the nth elementary
symmetric function. This will still be symmetric in the original summands and
so is a polynomial in AkE for k < mn. Again this is independent of the number
of summands provided that there are at least mn, and the resulting polynomial
is denoted Pm,n.
In section 2 we will calculate some of these polynomials Pn and Pm,n, giving
formulae when it is reasonable to write them down, and giving an effective
method to compute them in general. The key idea is that to understand
these operations that are not ring homomorphisms,we need to relate them to
5
operations which are. These operations are called the Adams operations, which
can be defined topologically, but are more easily described algebraically. In
our example of K°X, the Adams operation 4 k can be defined as the operation
which sends a sum of line bundles to the sum of their kth (tensorial) power. In
the theory of symmetric functions, the Adams operations are the power sums.
The power sums form a rational basis for the ring of symmetric functions.
This gives us a way to compute the polynomials P, and Pm,n, and is also the
starting point for the theory that we will use to describe K*(fQ°tmf; Zp).
The formula for the elementary symmetric functions in terms of power sums
gives us the statement that in a torsion free ring, the Adams operations determine the A operations. The torsion free condition reflects the need to cancel
coefficients. Bousfield generalized this theorem to describe Z/2-graded A-rings,
which are an algebraic model of K*X. The torsion free condition can be removed if we introduce p-local, or p-adic coefficients, at the expense of replacing
the Adams operation 4P (which is a ring homomorphism) with a non-additive
operation OP . The point is that once we introduce the coefficients, we only need
to worry about p-torsion. The operation 0P is related to 4P by the formula
P = x +po(x).
When we try to write AP in terms of the Adams operations, we have a formula
OP = pAp + -. and so OP is exactly what we want to get by cancelling the p.
If we have OPin advance then we do not need to worry about p-torsion.
Removing the torsion free requirement allows us to describe the p-adic Ktheory of an infinite loop space in terms of the p-adic K-theory of the corresponding spectrum. The idea is that with p-adic coefficients, we have stable
Adams operations Obk for k prime to p. Therefore if we introduce the operation
OP freely we should obtain a A-ring isomorphic to the K-theory of the infinite
loop space. In section 3 we give a detailed explanation of this theory which
we will apply in later sections.
We will apply this theory to the infinite loop space of the spectrum tmf. The
property of tmf that we will use is that for an even periodic spectrum R,
R A tmf can be identified with the stack of elliptic curves E together with an
GR. In section 4 we use this property to
isomorphism of formal groups E
describe the K-homology of tmf in terms of Katz's ring of divided congruences.
We can relate K-cohomology to K-homology using the Pontryagin duality
functor and so this gives us a description of the K-cohomology of tmf. We
expand upon this description in section 5 relating the K-cohomologyto the
Hecke algebra corresponding to the ring of divided congruences. We then use
Bousfield's theorem to obtain a description of the K-theory of QOOtmf.
6
2
A-rings
We begin with the definition of an abstract A-ring.
Definition 2.1 A A-structure on a commutative unital ring R is a sequence
of maps )Anfor n > 0 which satisfy the following conditions:
1. A°(r) = 1 for all r
R
2. A is the identity map
for n > 1
S. A(1) =
4. A"n(r+ s) = E=O Ak(r)An k(s) for all r, s E R
5. An(rs)= Pn(A'(r),...,An(r);A'(s),..., An(s))for allr,s
6. Am (An(r))= Pm,n(A'(r),...,Am n(r)) for all r
R
ER
where Pn and Pm,nare certain universal polynomials with integer coefficients.
We can think of any A-ring as a power series ring as follows. We make 1 +
R[[t]] into a A-ring by defining addition to be multiplication of power series,
multiplication by the formula
00
oo
00
(1 + E rntn)( + E Sntn) = 1 + E Pn(rl, ..., rn; 81, , Sn)tn
n=l
n=l
n=l
and the A operations by the formula
00
00
An(l+ E rmtm)= 1+ E P,n(rl,...,rmn)tm).
m=1
m=1
The fact that this does define a A-ring structure on 1 + R[[t]]is a consequence
of certain identities among the universal polynomials (see [Tal69]). The conditions in the definition combine to show that the total A-operation At(r) =
nl An(r)tn defines an injective A-homomorphism from R into 1 + R[[t]].
That is to say it is an identity preserving ring monomorphism which commutes with the A-operations. In particular the injectivity of the map follows
from condition 2.
7
2.1
The polynomials Pm,n and Pn
The universal polynomials Pm,n and Pn that occur in the definition of a A-ring
are somewhat mysterious. The polynomial Pm,n(el, ..., emn) can be described
xit),
([Tal69]) as the coefficient of t m in the product rIl<i<...<in<mn(1 +zi..
where ei is the ith elementary symmetric function in the indeterminates xj.
Similarly if fi is the ith elementary symmetric function in indeterminates yj
then Pn(e, ...,en; fl,...,fn) is the coefficient of t n in I,j(1 + xiyjt). These
descriptions arise when we consider the indeterminates xi, yi as universal line
bundles and the operation of taking exterior powers as A-operations. Specifically we write an element of R as (a pullback of) a sum of line bundles and then
the kth exterior power of this element is the kth elementary symmetric function
in the line bundles in the sum. An interesting problem is to give a more explicit
formula for these polynomials, directly in terms of the ei and fi. For example
since A' is the identity we find that Pl,n(el , ...,en) = Pn,l(el, ... , en) en. In
general however it is not true that Pm,n = Pn,m as A-operations do not commute. Below we give formulae in the next two simplest cases. In section 2.4 we
will discuss how to compute the formula in general and also how to compute
the polynomials Pn.
We begin with the following simple observation about binomial coefficients.
Lemma 2.1 For all m and p we have
P )
(e;l1)-=
Z_(_1)k(p m
_k) '
Proof
We prove this by induction on p, noting that for p = 0 the formula reads 1 = 1.
Now for p > 0 the binomial recurrence formula and induction give us
(m - 1
(r-i
=
=l
m
m1
P
r)(n
h))
(7)- ( (_1)k+lm ))
P
_k
k=O
El
8
Proposition 2.1 With the convention that e
=
1 we have for all n that
n
P2,n(el., e2n)=
(-1)k+len-ken+k
k=l1
Proof
We can represent a symmetric polynomial by looking at single monomial of
each type in the polynomial. For example x2x2 x3 and x l x2x4 have the same
type. In the product en-ken+k and P2 ,n each monomial that occurs has the
same type as x2 . x xi+l.. x 2 n-i for some i. To prove the proposition we
will compare the coefficient of this expression that occurs on each side of
the equation. The coefficient of x2 -... xi2xi+l... x2n-i in en-ken+k is zero if
n- k < i and (n(-k-i) otherwise. To see this we simply note that the terms in
en-k which can (and do) give this monomial must contain the indeterminates
xl through xi but then the remaining n- k- i indeterminates can be any
of the 2(n-
i) indeterminates xi+l through x2n-i. The polynomial P2 ,n is
obtained by taking the sum of all possible products of 2 distinct terms xs =
Xil ... xi, and Xj = Xj*--xjn. Each index set I such that {1,2,...,i C I
has a complementary set J such that xxj = X2... 2Xi+l ... X2ni . There are
(2(n-_)) ways to choose a suitable index set I. However this would give us the
term xIxj twice (once when we choose I and again when we choose J) so the
coefficient of x 2 ... xxi+l... x 2ni in P2n is (2(n-i)) = (2(n-it)- 1). Therefore
to prove the proposition it is sufficientto show that
(1k
2(n-i) A_=_ y d (_1)kk
2(n
-i) 18 =_ E(_1)k+1
12(n~-)
V,n-i-1 J
nk=l
2(n-i)
,n-kn-k-i J-i k=O---
'
This followsfrom the lemma with m = 2(n- i) and p = n- i- 1.
0
An alternative way to express the final calculation above is that
2(nk-i)
-1I, n-k-i
i
2 (-1)k+
= (1- 1)2(n - ) +
n-ii)/
i) to -1
The point here is that extending the sum to include k from -(ngives the same terms repeated, then adding k = 0 gives exactly the binomial
expansion of (1 - 1)2(n-i).
The case m = 3 is also sufficiently tractable for us to compute.
+ i be the
Proposition 2.2 Let w =-2 Ppio.Lecok+ cube root of unity. Then for all n
~'
P3,n(e, te3n)=
c?^1,k_eejek
lQ<k, +j+k=3n
9
wherec,s = 2R(wr) unless r = 0, t, 3t/2. If t > 0 then for r = 0, 3t/2 the value
of ct,, is whilefor r = t it is 2R(wr)+ (-l)t+l. If t = r = 0 the coefficient
is zero.
An alternative description of the coefficient of eleiek is that it is 1 if two of
1,j, k are equal and 2R (wl - ) otherwise, except when j = n when we must add
(-)n-l+1 if < n, and have zero for e. We defer the proof of this proposition
for the moment as a different approach will turn out to be much simpler, and
indeed give us a more useable formula in general. This formula was discovered
using similar reasoning to the m = 2 case and then a Maple program to
compute the coefficients. A key point in the calculation is that we can obtain
the nt h polynomial from the (n- )th by replacing every monomial elejek by
el+lej+lek+l and adding
0 c, ,3 nkeke3n-k. Indeed this description is easily
seen to be equivalent to the one given above and the missing terms in P3 ,1 give
the exceptions in the general case. The calculation reduces to a triangular
linear system for the last 3n/2 coefficients which is actually reasonable to
solve. In the appendix we give details of this calculation which involves many
relations between binomial coefficients.
2.2
Adams operations
The difficulty in calculating the universal polynomials is in large part due to
the fact that the A-operations are not ring homomorphisms. Indeed they are
are not even group homomorphisms. If we think of our indeterminates as
line bundles and elements of R as formal sums of line bundles then instead
of taking the elementary symmetric functions to get the A operations we can
take the power sums. This gives us the Adams operations 0p. To explain
this let us recall some facts about symmetric functions from [Mac79]. With
indeterminates x1 ,x 2,... we call the expression
en= ''
, . Xin
Xil
il <"'<in
the nth elementary symmetric function. Given a partition
= (
1 , Y2,...)
we
define e, = e,l e.2 .... Then the e as p runs over all partitions form a Z basis
for the ring of symmetric functions A. Equivalently A = Z[ei, e2 , ...].
Similarly the expression
Pn =
xi
is calledthe nth power sum. With p, definedsimilarlyto we obtain a Q basis
for A & Q. The power sums and elementary symmetric functions are related as
10
follows. For a partition p let mi = mi(/u) be the number of times that i occurs
in the partition. Let l(p) be the length of the partition (that is the number
of non-zero parts) and [I/ the sum of the parts. Define z = IJibl i"imi! and
u i1 =
)!.
Then
en = E (
)n-lO')
--1 zlP
IAI=n
) use =
Pn= n =(
(el,*--,en)-
]W=n
If we think of the indeterminates xi as universal line bundles and the exterior
powers as A operations then en corresponds to An and 'bn(xi) = xn. More
precisely we have the following theorem. ([Mac79])
Theorem 2.1 If R is a A-ring then for every x
homomorphism
q
:A
R there is a unique A-
R
which maps el to x. This homomorphism maps en to An(x) and pn to bn(x).
Proof
Once we define qx(el) = x then requiring ., to be a A-homomorphism forces
0$(en) = q$~(An(el))= An(x). This and the ring homomorphism requirement
defines ox uniquely on A. The latter statement is the definition of the Adams
n
operation.
Another way to express this result is that A is the free A-ring in the single
variable el. This theorem means that any formula in A is valid in any A-ring,
with appropriate translation. We can apply this to perform calculations for the
Adams operations. Firstly we find that the Adams operation can be expressed
as the Newton polynomial Vn(A1 , ..., An). This is the polynomial above which
expresses the power sums in terms of the elementary symmetric functions.
This polynomial can be calculated recursively from the relation
n
2 +...
= Alln- 1 - A2 4/n-
+ (-1)nAn- 1 4l + (-1)n+'nAn.
To prove this we note that in terms of the indeterminates xi this reads
zI
En =-zI
AdXi Ad Xi
-d
Xill- I
XA
±...
+
2+ * -
In the right hand side the jth term has monomials of lengths j and j + 1, for
j < n. The longer terms from the jth term exactly cancel the shorter ones from
the (j + l)th. For j = n we only have terms of length n which are cancelled by
11
the terms from the (n- 1)th term. After all the cancellation all that remains
on the right hand side is the terms of length 1 which is exactly the left hand
side.
Next we have the equations
,in
X + E yj)= ZX
= on (Z Xi)+Vpn(Zy)
+ E yjn
and
n(E Xi
(Xiyj)n
Yj)
=,On
(Zxi)+bn(Zyj)
which show that on is a ring homomorphism. Similarly
(on(EXi) = Om (EX') =
shows that
Ex
ZXn=
(EX)
= Vhmn.Also if p is a prime then the congruence (
bmebn
mod p shows that V(x)
xi)P
xP mod p.
This last property allows us to define a new operation OPon R by the formula
OP(x)
=- (
P
One can write down formulae for OP(x+ y) and OP(xy)just using the fact that
?,pis a ring homomorphism. It also follows that P(1) = 0 and pk0P = pvk
for all k. Conversely if we have an operation P which satisfies these formulae
then the operation bPdefined by the usual equation is a ring homomorphism.
2.3
The operation OP
The Adams operations make most calculations in a A-ring much simpler. In
this section we apply this principle to prove that if we have operations Ak
for k prime to p then by introducing the operation OPwith its usual formal
properties we obtain the Ak with k a multiple of p. To compute P(An ) we
first compute OP(An ) using the basis of power sums for the ring of symmetric
polynomials. The advantage of using the power sums instead of the elementary symmetric functions is that the Adams operations are described by the
formula V(pn) = pp, which follows easily from the fact that '/P is a ring
homomorphism. Combining this with the two inversion formulae gives us the
formula that we want. The formula for OP(An ) will then allow us to prove
Proposition 2.2.
Lemma 2.2 Modulo decomposables we have
op(An)- (1)(P-,)n,\pn
12
Proof
Working mod decomposables we have pa = (-1)n- l An and so applying 'P we
obtain
1(-1)nOP
(An)= Ppn= (-1)
1
pnAn
Therefore (-1)n-lpnOP(A n ) = (-l1)Pn-lpnAP" which gives the result in a torsion free ring. However this is a universal formula so it must hold in general.
[]
For p = 3 we can be more specific.
Proposition
2.3 Let w = e2 ri/3 and define cr,8 to be 2R(wr)
if r 7 0, s and 1
whenr =O, s. Then
03 (An) =
Cj-i,k-iAiAAk.
i+j+k=3n,iQ.'k
Proof
To compute 03 we first compute the Adams operation 03. We have two different formulae that we can use. The first comes from the Newton formula:
)3(An) = (n)
3
+ 3(A 3 (A n ) - Al(An),2(An))
The second formula comes from the formulae for elementary symmetric functions and power sums:
3 ( n) =
E
(-1)n-I(A)z.-lp3
IjA=n
/()
=
(-1)n-()zo
I[l=n
Z
II(
i>0
\4(A)
(-1)3i-z()
_3i
UL)
)
}v=3i
The first formula allows us to simplify the calculation of the second, based on
our knowledge of the universal polynomials Pm,n. Firstly it tells us that OP(An )
has degree 3. Secondly from the remarks in the appendix we see that all the
monomials of degree 3 are of the form Cl,j,kAl+l Aj+lAk+l where clj,kAlXjAkis a
monomial in ?p3(An-1). We should notice that the formula for the coefficients
in the theorem is consistent with this. The conclusion that we draw from this
is that in the second formula we only need to consider the terms which give
us monomials of degree 2 or less. The higher degree monomials must either
cancel out or be known. This means that in the second formula we can restrict
our partitions to have length no more than 2.
To perform this calculation let us first assume that n is odd. The partitions A
are of the form (k, n- k) or simply (n). From the definition we calculate that
13
Z(k,n-k) =
k(n - k) and Z(n) = n. Therefore from the formula for elementary
symmetric functions in terms of power sums we have
n
(-)--
o3(xn)=
n+ E
P+ ,
k=1
(T_
kP3(n-k)
k(n - k)' 3kp3(n-k)
ignoring terms of higher degree. Similarly from the formula for power sums
in terms of elementary symmetric functions we find that in Pm the terms of
degree 2 are
m-l
MA~m
/2)2'
2.j
L~
~~~
(-1)mkAm-k
+
L
m
2
k=l
where the last term is zero for m odd. We also have the degree 1 term
m(-1)m-lAm. Putting these expression together we find that in 'b 3 (An ) the
terms of degree less than 3 are
3n
-
'/1,
(3n(-).-,n. + 3n(l)3 .
Z AA.-k)+
L2.J
k=l
n
E k~n---k)°3k-1)3-k
~-k
k=1~
~ ~ ~~~~~~=
2
SA k
L
=
3) 3a -
3y
A;)a3 -k + 9
3
3(n-k)
k=l
k=1
If we divide by 3 we see that the coefficient of AkA3n- k in 03 (An) is 1 for k = 0,
-1 for k > 0 but not a multiple of 3, and -1 + 3 = 2 for k > 0 a multiple of
3. In other words it is k,3n-k. This completes the proof for n odd.
For n even the only difference is that the term P32n/
2 comes with a coefficient
1/2 which means that the coefficient C3n/2,3 n/ 2 of (3n/ 2 )2 is not 2 but 1 as
O
claimed.
As a corollary to this theorem we can prove proposition 2.2.
Proof of proposition 2.2
From the Newton formula and the calculation of A2 (An) we have
3 (n)
03 (n)
=
A'(An)A2 (An)
n
= An(](-1)k+lAn-kAn+k)
k=l
1 An(A2n)
+ ...
= (-1)n+
14
where we ignore the terms of degree 3. If we compare the formulae for the
coefficients in P3,,, and 3 (An) we see that the difference is indeed just the
(-1)n+lA(A 2 n) that we have here.
°
It is interesting to compare this proof to the combinatorial calculation given
in the appendix. The description of the coefficients as real parts of powers
of a cube root of unity, which is so crucial to the latter calculation, bears no
obvious relation to how they arise here. Indeed in the above we only need to
deal with real numbers.
2.4
The universal polynomials revisited
The Adams operations give us a much more effective way to compute the
universal polynomials P,n and Pn. Let us first consider the polynomial Pn
which gives An(rs) in terms of Ai(r) and Ai(s) for i < n. To compute this
we look at %bn(rs). On the one hand this is v(Al(rs), ..., An(rs)) while on the
other it is bn(r),O'(s) = vn(A(r),..., An(r))vn(Al(s),..., An(s)). Setting these
two expressions equal and rearranging allows us to write An(rs) in terms of
Ai(r), Ai(s), and Ai(rs) for j < n. This gives a recursive algorithm to compute
Pn for n as large as we please. In the appendix we tabulate values of Pn for
n < 10. A general formula would be difficult to write down but from the first
few cases we might conjecture that the sum of the coefficients is always zero,
for n > 1. This indeed is the case as we now show.
Lemma 2.3 The sum of the coefficientsin the nth Newton Polynomial v is
(_1 )n+l.
Proof
We can calculate the sum of the coefficients of vn(ei, -...,en) by evaluating
Ei= xTnat values of the indeterminates xi such that the ei all take the value
1. This means that -xi should be a root of 1 + x + x2 + . + xn for each i.
These roots, which we denote wi, are the (n + 1)th roots of unity, except for 1.
Then
Exi = (l)
n Elin
= (-l) n EW
1=
(-l)n Ewi = (-)n+l
since 1 + E wi = 0.
Theorem 2.2 The sum of the coefficients in Pn is zero for n > 1 and 1 for
n = 1.
15
Proof
The case n = 1 holds since P1 = elfi. Now suppose that the result holds for
all k < n for some n > 2. The equality
vn(\(rs), ..., An(rs))= n(Al(r),.
n(r))zn(Al()
An())
and the fact that the sum of the coefficients of a product of polynomials is the
product of the sums shows that the sum of the coefficients of the polynomial
vn(Al(rs),..., An(rs)) is ((-1)n+l) 2 = 1 for each n. However we also have
vn(Al(rs),..., A(rs))
A l(rs))
= Al(rs)nl(Al(rs)
l nAn(rs).
(-1)
Vn2(A(rs),,An-2(rs)) + --A2(rs)
By induction when we sum coefficients this reads
+-.-+(-)n+ln*x
1 = 1 * 1-0*1+0*1
where x is the unknown sum of coefficients in
An(r);A(s)
An(rs)= p (Al(r)
A(s))
[
Solving this gives x = 0 as claimed.
We now turn to the polynomials Pmn. In calculating P3,n we used the best
method available to us. That was firstly to calculate O'm(An) using the inversion formulae and then rearrange the Newton formula to calculate Am(An )
recursively. We actually fixed m but it is easier to fix n instead as this means
you are calculating what you will need for the next value of m. Again we have
tabulated some values in the appendix and can calculate the sum of the coefficients. In the calculations below we abuse notation and make no distinction
between an expression and the sum of the coefficients in that expression.
m (An) is 1 if m is odd and (-1)
Lemma 2.4 The sum of the coefficientsin Op
n
if m is even.
Proof
We have
n=
E (-1)nl()z-lp
IAI=n
so
Om (en) = E
(1)nI((J)z
lpmt -
Il=n
By the previous lemma Pn =
for m even
(-)
n+
l
so if m is even Pm = (-1) (1). Therefore
m(en) = (-1)n E
IIL=n
16
Z- 1.
If m is odd then Pm., = (-1)l( )- k where k is the number of odd parts of ,u.
Note that n- k is even, since mod 2, it is the sum of the even parts. Therefore
6(en)=
z
(_j)n-21(M)+kz-l
=
Z-1A
=
=~~~~~~
11,1=n
jpl=n
since n- 21() + k is even. To prove the lemma it remains to show that
ZI1I=n z 1 = 1. This is a classical result that goes back to Cauchy, though
perhaps a more accessible proof can be found in [Dwy38]
[
Theorem 2.3 The sum of the coefficients in Pm,nis 1 if n is odd or m = 1
and 0 for n even and m > 1.
Proof
We prove this by induction on m, the case m = 1 holding since A(A n ) = An .
To prove the induction step we use the newton formula again:
2 (An) +...
¥ m (An) = Al(An)?,m-l(An) - A2 (An)Obm-
(1)m+lmAm(An).
In n is even this reads
n ) + -.. + (- 1)m+lmAm(An)
1 = Al(An)- A2(AX
by the lemma so by induction we find that Am (An) = 0 for m > 1. If n is odd
it reads
1 = -A
(A n )
-
A2 (A)
-.
M A m (A n )
if m is odd and
-1 = A (An) + A2(AXn)
+ ...
mAm (An)
if m is even. In either case the equation gives Am(An)= 1 as required.
a]
2.5 K-Theory examples
The definition of a A-ring is set up so that for a CW-complex X, the ring
K°X has a A-ring structure with Akinduced by the operation of taking the kth
exterior power of a vector bundle. In this section we give explicit computations
in the cases X = S2, and X = BU to illustrate the operations Akand
P.
KoS2n
Let us recall from [HatOl] the ring structure on K 0 S 2 n. As an Abelian group
it is freely generated by 1, the trivial line bundle, and the external product
17
a = (H - 1) *
* (H - 1) of n copiesof the generatorH - 1 of K0 S2. Here
H is the canonical line bundle over S2 = CIP1. The multiplication is trivial
since (H - 1)2 = 0. If m denotes a trivial vector bundle of rank m then the kth
exterior power of m is a trivial vector bundle of rank (k) = m(m-1) ... (m-k+l)
Therefore on the first summand we have Ak(i) = (k) where the given formula
makes sense for all values of m. To calculate the Adams operations we note
that on a sum of line bundles L1 + .. + Lm, pk acts by taking it to the sum
Lk +.
+ Lk . A trivial line bundle raised to a power is still a trivial line bundle
so pk is the identity on trivial bundles. Finally this means that OP(m) = m-p
p
k~~~~k
To compute the A-operations on oa we first have to figure out the Adams
operations. The Adams operations are additive so once we know them on
vector bundles we can extend them to virtual bundles too. Once we know tpk
for all k we can define Ak on virtual bundles using the Newton formula. To
determine opk(O)we first compute that on S2 we have
k(H- 1) = Hk _ 1 = ((H-1) + )k - 1 = k(H-1)
where the last equality uses (H-
1)2 = 0. Then the equation
k(x) * pk(y)
ok(x * y) =
shows that o~k acts by multiplication by k on oa. Now we can show that
Ak(a) = (1)k+1kn-1a. Indeed this is true for k = 1 and if it is true for k then
the Newton formula shows that kna = ()k+lkAk(oa) mod terms of degree 2.
The latter all vanish by the induction hypothesis since a2 = 0 and so dividing
by k gives the result. The formula for 4 P(a) also gives that OP(a)= pn-1a.
This calculation illustrates two important points. The first is that if we want
to determine a A-ring structure then we may well have to find the Adams
operations first. The second is that if we can do this, then so long as the
ring is torsion free (to allow us to cancel the k) we will have determined the
A-operations. This latter statement is the content of the following theorem of
Wilkerson.
Theorem 2.4 If R is a torsion free unital commutative ring equippedwith
ring endomorphisms ?/k : R -+ R for k > 1 such that
1. ?P1 = Id
2 Vpkm
=
km
3. P(r) _ rP mod pR for p prime
then R has a unique A-ring structure with the given
18
pk
as Adams operations.
KOBU
The space BU is a topological direct limit of spaces BU(n). The space BU(n)
is the classifying space for complex vector bundles of rank n. In can be constructed as the complex Grassmannian Gn(C ° ° ) of all n-planes in C'. Atiyah
and Hirzebruch proved that as a ring K°BU(n) can be computed as the completed representation ring R(U(n)) which itself is the ring of invariants in
R(T(n)) under the action of the Weyl group, where T(n) is the maximal torus
of diagonal matrices. Since R(T(n)) = Z[[zl, ..., zn]] and the Weylgroup is the
full symmetric group permuting the variables we find that R(U(n)) is also a
power series ring in n variables, which are the elementary symmetric polynomials in the zi. Taking a limit we find that K ° BU is the power series ring in
infinitely many variables.
The proof of the theorem gives a useful description of the variables in K°BU(n).
The cohomology of Gn(Co) is shown in [HatO1] to be a polynomial ring in
the Chern classes cl,...,cn of the universal bundles over Gn(Coo). The oper-
ation of taking the ith Chern class of a vector bundle extends to operations
Ci: Ko°(X) -- H 2i(X) which behave additively much the same as A-operations.
As before we consider the ring homomorphism that results by looking at power
sums instead of elementary symmetric functions and we obtain the Chern character ch = Zk chk = Ek Vk(C1, ... , ck)/k!. Formally we can say that if L is a
line bundle then ch(L) = e(L). The Chern character gives a map to the direct
product of the individual cohomology groups rather than the direct sum, which
in the case of Gn(Co) means power series in the ci rather than polynomials.
The key theorem is that if X is a CW-complex with torsion free homology
then
ch: k°(X) - H*(X) Q
is injective. In the case of X = BU we have one element i E K°(BU) given
by the identity map of BU and using the exterior power operations we obtain
a family of elements A(i).
Theorem 2.5 The elements ch()An(i))E H*(BU)0Q form a set of polynomial
generators and so K°(BU) = Z[[A(i), A2 (i)...]].
Proof
We give a sketch and refer to [Hir61] for details. The idea is to introduce the
total y-operation defined by yt(x) = At/-t(x). The -operations ?i, defined
by yt(x) = E Yi(x)ti,give another set of generators for the polynomials in the
Ai , so it is sufficient to prove that the Chern characters of the 'ii are a set of
generators. If we have At(xi) = (1 + xit) then ch(At(xi)) = (1 + exit) and so if
we introduce new indeterminates zi = xi - 1 we have
ch(yt(zi)) = (1 + (exi - 1)t).
19
Now ei -1 = xi + decomposables so any polynomial in the xi can be written
as a polynomial in ex - 1 plus higher terms. To prove the theorem we start
with a maximal torus T(n) in U(n) and consider the product B2k of n copies
of CPk. The cohomology of this product is a polynomial ring in n variables
xi modulo terms of degree k + 1 and using the Chern character we find that
K ° B 2 k is the same truncated polynomial ring in the zi. Taking an inverse limnit
over k we obtain that K°BT(n) is a power series ring in n variables. Next a
naturality argument gives us K°BU(n) and another inverse limit gets us to
K*BU
°
The power series ring K°BU is a completed version of the free A-ring in one
variable. In fact it is the inverse limit over n of the quotient by the ideal
generated by terms of degree at least n. This means that the actions of Ak
and 9P are given by the universal formulae described above.
The y-operations introduced in the proof of theorem 2.5 give rise to the so
called -y-filtrationI = '(I) D 2 (I) D) _... of a A-idealI. Here Ln(I) is the
ideal generated by products yil(xl)... Yik(xk) where il + ..-- + ik > n and
xi E I. I is called y-nilpotent if it is nilpotent and yi(x)= 0 for x E I and i
sufficiently large. We remark that in the above calculation the augmentation
ideal of K*B2k is y-nilpotent, the operations Yibeing zero for i > n, and any
polynomial with zero constant term raised to the (k + 1)th power being zero.
3
Bousfield's Theorem
In the previous section we saw how in a torsion free unital commutative ring,
a A-ring structure is equivalent to a set of Adams operations. We also saw how
in K-theory the operation of taking the kth power of a line bundle gives rise to
Adams operations and hence a )A-ringstructure on the K-theory of a space. A
natural question to ask is whether the K-theory of a spectrum E is also a Aring. The answer is generally no because the Adams operation pk only defines
a stable operation if k is a unit in the coefficient ring. If we were to work over
the p-adics for p a prime then we would have all the Adams operations save
for those which are a multiple of p, plus we wouldn't have to worry about the
ring being torsion free. In this case by formally introducing P, or indeed as
we saw previously even P , we would expect to obtain a A-ring. We would also
expect this A-ring to be closely related to K°(Qf'E; Zp). This is the content of
Bousfield's theorem. The object of this section is to understand the theorem
in detail so that we may apply it to the spectrum tmf and give explicit details.
There are two main issues in stating the correct theorem. The first is determining the formal algebraic properties that characterise K*(X; Zp) and K*(E; Zp)
for a space X and spectrum E. This involves the notions (in the language of
20
Bousfield) of Z/2-graded p-adic A-rings and Z/2-graded p-adic 0p-modules. The
second issue is establishing which properties will be required so that the introduction of the P operation to K*(E; Zp) gives a unique (p-adic) A-ring structure. The first of these conditions is that K*(E; Zp) should be torsion free, due
to this being the class of spectra that can be constructed starting from E = K.
The second of these conditions is more subtle and relates to the -filtration
of the augmentation ideal (K°,K'). The operations Ok must satisfy the relations k(y) = ky mod r2( K0 (E; Zp)) for y E K°(E; Zp) and pk(x) = x
mod F2 (Kl(E; p)) for x
K(E; Zp). If K*(E; p)/F 2 (K*(E; p)) = 0
then these conditions are automatic and the operation pP, or equivalently
OP, can be introduced freely to give K*(Q°OE;Zp). Otherwise these conditions must be included as part of the data when we introduce P . The space
K*(E; Zp)/r2 ( K*(E; Zp)) can be identified as the first and second stable cohomology groups of E which gives a more useable criterion for K*((QE; Zp)
to be a free object.
3.1
Z/2-graded p-adic A-rings and Z/2-graded p-adic modules.
If X is a connected CW-complex then by considering the finite subcomplexes
of X and regarding Zp as a (topological inverse) limit of finite p-torsion groups
we can write K*(X; Zp) as a limit of rings R with the following properties:
1. R is an augmented, Z/2-graded commutative ring.
2. R° is a A-ring and R 1 has endomorphisms O'~for n > 1 such that Al =
Id, opJopk = pjk and Opk(xy) = k(x)Opk(y) for x
R and y
R.
FurthermoreAk(xy)= (-l1)fn-Jiii (x)i (y)... Vi (x)qii (y) for x,y E
R 1 where the sum is over all strictly increasing partitions (il, ... , ij) of k.
3. The augmentation ideal R = (0, R') is finite p-torsion and 'y-nilpotent.
An inverse limit of such rings R is called a Z/2-graded p-adic A-ring.
A ring R satisfying properties 1,2 and 3 also has the following properties:
4. Fr2 (R) = {x E Rl(OP)nx= 0 for some n > 0}.
2 (R
5. OP(y)= y mod rP
6.
7.
ky =
OpkX
ky mod
x
)
2 (R0 )
for y ER0 .
for y E
AO.
mod P 2 (R l ) for x G R1.
21
8. The Adams operations
R satisfy
k:
= ikpk+p"for k > 0 and
somer.
9. pk(xy) = kpk(x)bk(y)
for x, y E R 1.
There are several remarks to be made here. Firstly R' gives rise to a A-ring
Z E R1 with trivial multiplication, where (-1)k+lpk corresponds to Ak. Then
BPis identified with 9Pso that F2 (R 1) = {x R (P)nx = 0 for some n > 0}
and property 7 is consistent with property 5. We note that to obtain the
A-ring corresponding to the group R we apply the functor which is the left
adjoint to the forgetful functor from the category of A-rings to the category
of abelian groups with endomorphisms V)n. The maps in the latter category
are homomorphisms that commute with the opn. Secondly the operations ibk
on R can be thought of as pk/k as in [Bou96] p17. In this case the forR 1 which is
mula in property 9 becomes pk(xy) = k(x),pk(y) for x,y
consistent with [GoeO4]. We note that this formula comes from the fact that
the Adams operations commute with the Bott periodicity isomorphism only
up to multiplying by k. Finally we remark that the formula for An(xy) with
x, y E R 1 is seen to be equivalent to property 9 when we consider the formula
en = El'l=n(-1)n-l()z;lp,
and remark that since squares are zero in R', we
only need to look at the partitions I with no multiplicities greater than 1.
If we ignore the operations /)k for k prime to p then we arrive at the definition
of a Z/2-graded p-adic OP-ring. Specifically it is an inverse limit of rings R
over Zp with operations P : R - R° and Opb:R 1 -+ R1 , such that 9p defined
by the usual formula is a ring homomorphism on R ° , and VP on R 1 is a group
homomorphism. With the convention 4 p = 0P on R1 in 4 and 5 as remarked
above, R must satisfy properties 1, 3 and 5. We don't have the -- operations so
property 4 is used here as a definition and -y-nilpotent reduces to just nilpotent.
Furthermore the operations must be consistent with properties 2 and 9, which
means we have the formulae ,P(xy) = 'P(x) 9 (y) for x E R° and y E R1 and
OP(xy) = pP(x)P(y) for x, y E R 1 .
Bousfield proves that if we take a Z/2-graded p-adic 0P-ring and endow it
with Adams operations Ok for k prime to p that satisfy properties 7,8 and 9
above, together with the usual properties described in section 2.2, then the ring
has a unique structure as a Z/2-graded p-adic A-ring with the given Adams
operations. We remark that property 2 is straightforward so the difficulty here
is in proving that /i vanishes on the augmentation ideal for sufficiently large
i, so that R is 7y-nilpotent.
This theorem can be viewed the other way around too. If we consider K*(E; Zp)
for E a spectrum, then additively we have a limit of Z/2-graded finite abelian
p-groups which are equipped with Adams operations V)kfor k prime to p. This
= ,)jk .
means that ?bksatisfy property 8 plus the conditions b1 = Id and VPJVbk
22
A space which satisfies these properties is called a Z/2-graded p-adic Vp-module.
These properties mean that the action of Z+ \ pZ extends to a continuous action of the p-adic units Zp. We can make such a space into a Z/2-graded
p-adic P-ring by simply applying a functor which forces conditions 1,3 and 5
to be satisfied and conditions 7 and 9 will naturally be satisfied too. Bous-
field's theorem is that when we apply this functor to K*(E; Zp), at least when
it is torsion free, then we will get K*(QE; Zp), where the subscript 0 denotes
the baseloop component of the loopspace.
3.2
The free SP-ring functors
Bousfield constructs (Z/2-graded p-adic) P-rings out of p-profinite abelian
groups using the adjoint functor method for various forgetful functors. To
motivate the definitions let us consider the case when our abelian group has
Adams operations, so that when we make it into a OP-ringit becomes a A-ring.
In a A-ringR the ideal P2((R) is a A-idealso we can consider a decomposition
R= z + r(R) +R/p 2 (R)
which is preserved by the A-operations. The summands are also closed under
addition and multiplication but the sum is not necessarily direct. In the p-adic
case this decomposition is a limit over aeof decompositions
R= z + F2(Ra)+Ra/P2 (Ra)
= 0 for some n > 0} and 0P acts as the iden= {x E
where p2(,)
tity on R/F 2(R ). This means that these decompositions can be described
without reference to the Adams operations Ok for k prime to p. If we forget
-l(oP)nx
the operational structure but remember the decomposition then we should be
able to recover the original A-ring from this. Encoding the decomposition as
the map
- R/r 2 (R) gives us the functor W. This can be built up out
of functors T and J that are left adjoint to functors which remember R and
R/p 2 (R) respectively. T is the simplest way to introduce 0P freely but it forgets the decomposition of R so it doesn't give us the A-ring that we want
unless p2 (R) = R. Restricting (the right adjoint of) T to rings which satisfy
this condition gives us the functor T, which is the special case W(M -- 0).
Similarly if it happened that Pr2 (R) = 0 then the functor J would give us what
we want and so J is the special case W(0 c H).
We recall the definitions of T, J, W and T from [Bou96]
T is left adjoint to the functor sending a Z/2-graded p-adic 0-ring R to the
Z/2-graded p-profinite abelian group R.
23
J is left adjoint to the functor sending a Z/2-graded p-adic OP-ringR to the
Z/2-graded p-profinite abelian group R/p 2 R. Here p 2 R is defined as the limit
of p 2Ra where R = lim Ro.
W is left adjoint to the functor sending a Z/2-graded p-adic 0P-ring R to the
map of Z/2-graded p-profinite abelian groups R
-+
R/r 2R.
T is left adjoint to the functor sending a connective Z/2-graded p-adic Or-ring
R to the Z/2-graded p-profinite abelian group R. Here connectivemeans that
/r 2 R= 0.
To evaluate these functors on specific groups we can use the fact that left
adjoints preserve cokernels to reduce ourselves to dealing with free groups.
The functors J and T are simple to describe. In the case of T we note that a
connective Z/2-graded p-adic Or-ring R can be written as R = Zp ff r 2 R and
the adjunction relation is
r 2R).
Homep(TH,Zp· r 2R) = HomAb(H,
In the case where H consists of Zp on an even generator x and an odd generator
y this formula tells us that
T(Zp(x), zp(y))
= zp[[x, Opx, (9p) 2 x,
...]]6Ap[[y,
y, (0p)
2
y,...]].
To find J we use the identifications ([Bou96]) Ro/ 2 R 0 - {u e RI°loPu= O}
and R 1 /r 2 R'
-
{u G R1 kpu= u} and obtain
J(H) = Zp[[H]]6Ap[[H1]]
where 0p acts by zero on H ° and OP is the identity on H 1 . Note that we can
view this functor as starting with T then introducing by the relations that
exist in R/F 2 R.
H) =
Next if we write down what the definitions entail we find that W(M
T(M)6J(M)J(H) and so we can compute W from T and J. Specifically, the
action of OPon a tensor product is given by the formula
OP(r0 s) = O(r) 0 sP + rP0 O(s) +poP(r)0 OP(s)
which extends the action on the two factors. In this case the tensor product
is over J(M) which means that we introduce relations given by equalizing
the actions on the two factors. The action of J(M) on J(H) is given by the
natural map J(M -- H). The action of J(M) on T(M) is also given by a map
J(M) -- T(M). M is naturally a subset of T(M) and this map extends the
inclusion to J(M), whose elements are formal 4_combinations of elements of
M. Another way to describe this map is as the adjoint of the map obtained
24
by composing the adjoint of the identity of T(M) (which is the inclusion of M
in T(M)) with the projection to T(M)/F 2 T(M).
This just leaves us needing to describe T. We can describe T(Zp, 0) as follows.
of Zp[x, 0Px, (P) 2 x, ...]
Let Tp(Z/pm ) be the cokernel of the OP-endomorphism
which sends x to pmx. Since OP(pmx) = pmOPxmod decomposables, this is a
polynomial ring with coefficients in Z/p m , though with a more complicated OP
T
operation. Let I C Tp(Z/p m ) denote the OP-idealgenerated by (Tp(Z/pm))n
together with (OP)n(OPx- X) for x G Tp(Z/pm). Then Tp(Z/p m )/I, is a p-adic
OP-ring and we have
T(Zp, O)= limmlimnT(Z/pm )/In.
As a ring this is a power series ring in infinitely many generators as with T,
but the 9P operation is more complicated.
The functor T is not the easiest to to understand so a reasonable question to
ask, given the decomposition of R, is whether we can build W out of J and
the functor To which is left adjoint the functor sending R to r2R. In fact
since 2R = {x E Rl(OP)nx = 0 for some n > O} in the finite case, we see
that OP-maps preserve these spaces and so To(M) is simply the connective P
ring T(M) viewed as a non-connective ring. Intuitively the functor sending
R
R/p 2 R to T0 (r2 R)®J(R/rP2 R) will be W provided that we know how
2 R. Specificallywe have the following
to reconstruct R from P2 R?and R/rP
proposition.
Proposition
3.1 Let C be the category of Z/2-graded p-adic OP-ringsR to-
gether with a splitting R = P2 (R)
R/P2 (R) that is preserved by the mor-
If Wo is the left adjoint to the functor sending R E C to the split
surjection R - R/rF2 (R) then Wo(M E H - H) = To(M)J(H).
phisms.
Proof
This is just an exercise in definition chasing.
l
This proposition is a generalisation of the case H = 0 which gives the functor
T. However the naturality condition seems unlikely to be satisfied which limits
its usefulness.
3.3
K*(Qo E; Zp)
The free OP-ringfunctors above are defined on the category of Z/2-graded pprofinite groups. When we come to describe K*(Qo E; Zp) we will apply them
to Z/2-graded p-adic 0h-modules to obtain Z/2-graded p-adic A-rings. More
25
precisely let A-ring, -mod, P-ring and Ab denote respectively the categories
of Z/2-graded p-adic A-rings, /2-graded p-adic O-modules, Z/2-graded padic P-rings and Z/2-graded p-profinite abelian groups. Then we have a
commutative diagram of forgetful functors
A-ring
,
+-mod
I
OP-ring
I
Ab
-
where the horizontal arrows are the various functors defined above and the
vertical arrows keep the underlying space but forget the operational structure.
These diagrams correspond to a commutative diagram of adjoints
A-ring
+-mod
-
I
TI
Ab
OP-ring
where the functor I introduces the operations 4 k for k prime to p, and is in
fact the same functor in both vertical arrows. The functors referred to below
are the top arrows in this diagram, but in fact they are the same functors
as the bottom arrows defined in the previous section. This is because the bk
commute with OP and so introducing 0P after the Ok is the same operation as
introducing P before the Ok.
We are now in a position to state Bousfield's theorem. Let E be a spectrum
with O-connected section E(O) and consider the Atiyah-Hirzebruch spectral
sequence for K*(E(0); Zp). The picture for E2 is shown below.
t
2
0
1
0
0
0
-1
H1
. H2
H1
H2
H3
H2
H3
0
0
1
s
Here every group to the left of the picture is zero since we take the zero
connected section of E. This means that there are no differentials that hit the
groups Hi(E(O); Zp) and H 2 (E(0); p) and so there is an edge map
K*(E; Zp)H K*(E; Zp) - K*(E(0); Zp) -+ {H 2 (E(0);Zp), H'(E(0); Zp)}.
26
We remark that this is a quotient map and so is surjective.
Theorem 3.1 If E is a spectrum with K*(E; Zp) torsion free then
K*(QoE; Zp) - W(K*(E; Zp)H)
as Z/2-graded p-adic AP-rings. In particular if E is also O-connectedwith
H 2 (E; Zp) = H'(E; Zp) = 0 then K*(Q°°E; Zp) T(K*(E;Zp)).
Both statements in the theorem are consequencesof the followingproposition.
(The former requiring a bar spectral sequenceargument while the latter being
trivial.)
Proposition 3.2 If E is a spectrum with K*(E; Zp) torsion free then
K*(Q°°E(2);Zp) - T(K*(E; Zp)).
Proof
The proof of this statement is to check that it works for E = K and E = EK
and then build up the class of spectra with torsion free K-theory from these.
Let us consider the case of E = K. According to [Tor77]K' (K; Zp) = 0 and
K°(K; Zp) is the completed group ring Zp[[Zp]]. Here Zp is the group of units
in Zp and a E Zp corresponds to the stable Adams operation ?/i. This means
that as a Z/2-graded p-adic p-module K*(K; Zp) is a free module on the single even generator x corresponding to the identity map. When we introduce
OPby applying T we therefore obtain the free Z/2-graded p-adic A-rings on
an even generator, which is Zp[[x, A2 x, A3 X, ...]]. The 2-connected cover of K
has zero space BSU, which is the 2-connected cover of BU. The other spaces
in K(2) are given by the homotopy equivalences QBSU _ SU, QSU _ BU,
QBU
U and QU
Z x BU. The calculation of K*BSU is similar to
that of K*BU is section 2.5. The cohomology of BSU is again a polynomial ring in infinitely many chern classes, with c omitted. The conclusion is
that K° (BSU; Zp) Z[[y, A2 y, A3 y, .. ]], where y corresponds to the covering
BSU -+ BU. Therefore we see that in this case K*(BSU; Zp) - T(K*(K; Zp))
which is what we want. For E = EK the indices are shifted so now K*(EK; Zp)
is a free module on the single odd generator x. Now we get that T(K*(EK; Zp))
is the free Z/2-graded p-adic A-rings on an odd generator, which is the completed exterior algebra Ap[[x,A2 X, A3 x, ... ]]. Shifting indices in the 2-connected
cover we have the zero space SU. The calculation of K*U can be found in
[Ati67] and with suitable modifications for SU, we have that K'(SU; Zp) is
also a completed exterior algebra on infinitely many generators.
27
3.4
Examples
In the previous section we saw how Bousfield's theorem applies to the the
spectra K(2) and EK(2). Now we look at some other examples to help us to
understand the theorem further.
Given a spectrum E we will want to compute the stable cohomology groups
H2 (E(0); Zp) and Hi(E(0); Zp) of the 0-connected cover of E. If we know the
homotopy groups r E and ir2 E then we can use the following theorem.
Theorem 3.2 If E is a CW-spectrum and G is any coefficientgroup then
HI(E(0))- -irE
H2 (E(0)) - coker(x7: rlE --* r2E)
and so
H'(E(O); G) ~ Hom(7rE, G)
H2(E(0); G) - Hom(H2(E(0)), G) (3 Ext(irlE, G).
Proof
The formula for H1 (E(0)) follows from the Hurewicz isomorphism. To compute
H2 (E(0)) consider the cofibration
E(1)
E(O) -
Hr 1,E
where HirlE is the Eilenberg-Maclane spectrum. The stable homology groups
Hn+l(K(II, n)) and Hn+2(K(II, n)) are computed in [Mac54] as 0 and II 0
Z/2 respectively. Therefore part of the long exact homology sequence for the
fibration is
7rE
0 Z/2 7 r2 E
H2 (E(0))
HI(HirlE) = 0
where the second group is given by the Hurewicz isomorphism for E(1) and 7
is the non-zero element of 7r. This sequence gives the formula for H 2 (E(O)).
The formulae for the cohomology groups follow from the universal coefficient
0
theorem.
We remark that in the preceding proof we could equally well have used the
long exact sequence in cohomology. The stable cohomology groups we would
need ae Hn(K(II, n); G) = Hom(II, G), Hn+ (K(l, n); G) = Ext(II, G) and
Hn+ 2 (K(II, n); G) = Hom(II 0 Z/2, G). The cohomology long exact sequence
then gives us isomorphisms Hi(E(0); G) - Hom(7rE, G) and H2 (E(0); G)
28
(Xr))*
ker (Hom(-7r
2 E, G) (*
Hom(7r1E 0 Z/2, C)) ®Ext(7rlE, C) which amounts to
the same as above.
We can use this theorem to look at some examples. Let us begin with the
spectrum K. This has homotopy groups 7r2 = Z and 7r2n+1= 0. This
means that H 1 (K(O);Zp) = 0 and H 2 (K();p)
= Zp. The space POOKis
x BU so the connected component of the base loop is just BU. We saw
that K°BU is the free Z/2-graded p-adic A-ring on an even generator, while
K°(K; Z)
Zr[[Zp]. Therefore we have the formula
W(Zp[[,
]] __+ZP)
ZP[[X, A2(X),...]].
A second example is the sphere spectrum S ° . The K-cohomology of this
spectrum is simply the coefficient ring Zp in degree zero. The groups r So and
7r2 S ° are well known to be Z/2 generated by q and 772 respectively. From the
formulae in theorem 3.2 this gives us HlS°(O) = Z/2 and H2 S°(0) = 0 and
so we have H'(S-(0);p) = 0 and H2 (S0 (O);Zp)= Ext(Z/2,Zp). This last
group is zero for p > 2 and Z/2 for p = 2. The space Q'S ° is what is usually
called QSg. The integral cohomology of this space was calculated by [Pri70]
as the cohomology of the classifying space of the infinite symmetric group,
BSoo. This isomorphism is induced by a map so we can apply the AtiyahHirzebruch spectral sequence to deduce that the K-cohomology of QSo is also
isomorphic to the K-cohomology of BSo. This latter ring can be calculated
using Atiyah's theorem that for a finite group G, K*BG is isomorphic to
the completed representation ring R(G). Atiyah proves that the dual of the
representation ring of Sn is the ring of symmetric functions of degree n in n
variables. Taking a direct sum over n gives the polynomial ring Z[cl, c2 , ...] in
the elementary symmetric functions. Dualising and completing we have the
free Z/2-graded p-adic OP-ringon an even generator. Therefore in this example
the theorem gives us the formulae
W(Zp
-
0) = T(Zp)
OPx...]]
OZp[[x,
for p > 2 and
W(Z 2
-+
Z/2)
Z 2 [[x, 9x, ...]].
Note that these rings, as in the example above, are power series rings in infinitely many generators. The difference is in the grading of the generators:
the operation A increases the grading by i while 0P increases it by p.
More generally we could consider the Moore spectrum MA for an Abelian
group A. The cohomology of this spectrum (by definition) consists of A in
dimension zero only. Therefore the Atiyah-Hirzebruch spectral sequence is
trivial and gives us that K°MA = A also. This means that to apply Bousfield's
theorem we have to assume that A 0 ,p is a torsion free group, or equivalently
29
that A has no p-torsion. The homotopy groups of MA for torsion free A can
be found using the following lemma.
Lemma 3.1 If A is a torsion free abeliangroup then rnMA - A 0 lrnS° .
Proof
Let
O
k F2
be a free resolution of A, which occurs as the homology of a cofibration sequence
Vs°
--+ Vs° -
MA.
In fact we would construct MA as the cofibre of a map between a wedge of
spheres like this. The long exact sequence of homotopy groups has terms
-*F 1
0 irnS°
l
F2 0 irS °
-
rnMA
and so can be broken up into short exact sequences
0 -+ coker(k ® 1) - xrMA
ker(k
1)
0.
To identify the terms in this sequence we can tensor our free resolution of A
with 7rkS° and obtain an exact sequence
F1
°
7rkS
F2
rkS ° -- A
rkS ° -- 0.
This tells us that the cokernel in the short exact sequence above is A 0 rnS °
and the kernel (by definition of Tor) is Tor(A, r, _lS°). Since we are assuming
that A is torsion free this Tor term vanishes and the short exact sequences
become isomorphisms rnMA _--A 0 7rnS°.
[
From this lemma and the previous proposition we deduce the isomorphisms
Hom(A 0 /2,Zp) = 0 and H 2(MA(O);Zp) _ Ext(A
Z/2, Zp) for torsion free A. In fact these results still hold under the weaker
assumption that A has no p-torsion, or indeed with no assumptions on A.
H'(MA(O);Z)
Lemma 3.2 If A is an Abelian group then
H1(MA(O);Zp) - Hom(A0 Z/2, Zp)= 0
and
H2 (MA(O); Zp)
Ext(A 0 Z/2, Zp) - Hom(A, Z/2 0 Zp).
30
Proof
The result for H 1 is clear since the tor term vanishes when n = 1. For n = 2 we
find that H2MA (0) - coker(xr)
-
r2 MA/(A 0 Z/2) - Tor(Z/2, A)
2 (A)
H 2 MA
vanishes here too. The final isomorphism can
so that the Hom term in
be obtained using the following result from [CE99]:
If C is torsion free and B is a torsion module then the sequence
O-C - Q C - Q/z C--O
is exact and so
Hom(B,Q/Z 0 C) - Ext(B, C).
In our case B is the torsion module A 0 Z/2 and C is the p-adics. There-
fore we have isomorphisms Ext(A 0 Z/2, Zp) - Hom(A ® Z/2, Q/Z 0 Zp) Hom(A, Hom(Z/2, Q/Z Zp)) - Hom(A, 2(Q/Z) 0 p) -- Hom(A,Z/2 ®ZZp).
We now state our conclusion in the following proposition.
Proposition 3.3 Let p be a prime and let A be a abelian group that has no
p-torsion. If p > 2 or p = 2 and A = 2A then K°(Q'°MA; Zp) is the free
p-adic OP-ringon A Zp. If A/2A $ 0 then K°(QO°MA;Z 2 ) - W(A 0 Z2
Hom(A/2A, Z/2)).
Proof
The proposition follows from Bousfield's theorem and the isomorphisms
Hom(A,Z/2 0 Z2) - Hom(A,Z/2) - Hom(A, Hom(Z/2, Z/2))
-Hom(A 0 Z/2, Z/2)
for p = 2. (For p > 2 the first term is of course zero.)
O
In the subsequent sections we will apply Bousfield's theorem to the example
E = tmf to obtain a description of K*(Qo tmf; Zp). For the moment let us
record the computation of H' (tmf (0); Zp) and H 2 (tmf (0); Zp).
Lemma 3.3 For all primes p
H'(tmf (0); Zp) = 0.
For p > 2, H 2 (tmf (0); Zp) = 0 while
H 2(tMf (0);Z2 )-
Z/2.
Proof
The homotopy groups of tmf agree with those of the sphere spectrum in small
O
dimensions so the result is the same as for S ° .
This result means that at odd primes K*(Qotmf;
31
Zp) is a free OP-ring.
4
The K-homology of tmf
4.1
K-cohomology and K-homology
In the previous section we described Bousfield's theorem which gives the Z/2graded p-adic A-ring K*(Qo E; Zp) functorially in terms of K*(E; Zp) when the
latter is torsion free. Our next objective is to apply the theorem to the case
E = tmf. To describe the K-cohomology of the spectrum tmf it will be convenient first to consider K-homology. To this end we now describe the relation
between K-homology and K-cohomology with various coefficients. Let Zpoo
denote the p-torsion subgroup of Q/Z. Also for a locally compact Hausdorff
abelian group A let A# be the Pontryagin dual of continuous characters of A
into S 1. (Note that if A is finite then such characters must map into Q/Z.)
The relevant result is the following (corollary 2.3 of [Bou99])
Proposition 4.1 For a space or spectrumX there are natural isomorphisms
Kn(X; ZP)
Kn(X; Zpoo)#
Kn(X; Z/p)
K. (X; Z/pi)#
fori, n E Z with i > O.
Proof
To prove this statement we simply need to check that the expressions on the
right hand side define a cohomology theory in X and that the coefficients agree
(naturally). The first statement follows from the fact that the Pontryagin
duality functor is exact and transforms direct sums into direct products. To
see exactness it suffices to show that S1 = R/Z is injective. To prove the
injectivity of R/Z we use Baer's criterion, which says that for a ring R, an R
module M is injective if and only if every R module homomorphism from an
ideal of R into M, extends to all of R. In our case R =
so an ideal of R
is of the form nZ for some n. If f: nZ -+ R/Z is a homomorphism then we
can find some x R/Z such that nx = f(n). We can extend f to all of by
defining f(1) = x. To see the preservation of limits we simply note that the
Pontryagin duality functor is representable and hence continuous. It remains
to check that the coefficients agree. In the first case this means we want an
isomorphism Zp - Hom(Zpoo,R/Z) - Hom(poo, Zpoo). The map sending a padic number to the map multiplication by a gives us such an isomorphism.
In fact since this map is induced by the multiplication map of the spectra KZp
and KZpoo this gives us the naturality that we want. The argument in the
second case is the same.
[
32
As a corollary to this proposition we find that if Kn(X; Zp) is torsion free then
Kn (X; Z/p) - (Kn (X; Zp)/pKn(X; Zp))#. In addition to the functorial de-
scription of K*(Q E; Zp), Bousfield proves that is torsion free when K*(E; Zp)
and so there is the following statement in terms of mod p K-homology.
Theorem 4.1 If E is a spectrum with K*(E; Zp) torsion free then there is a
Z/2 gradedHopf algebraisomorphism
K*(QoE;Z/p) - (W(K*(E; Zp)H)/pW(K*(E;
Zp)i))#
The K-homology groups of an infinite loopspace, with mod p coefficients, are
an attractive object to study, as they are a commutative cocommutative Hopf
algebra. These objects can be understood quite well. For example in the case
where E is 2-connected we find that K*(Q'E; Z/p) is a tensor product of Witt
and exterior Hopf algebras.
4.2
The ring of divided congruences
According to proposition 4.1 we can recover the K-cohomology of a spectrum
with Zp coefficients from the K-homology with coefficients in Zp. We will
see below that we can describe the K-homology of tmf in terms of Katz ring
of divided congruences. We describe this ring now. We recall some definitions
from [Gou88] Let N be an integer prime to p and A be a p-adic ring (meaning
that A is a Zp-algebra which is complete and Hausdorff in the p-adic topology).
If E is an elliptic curve over A then a level N structure is an inclusion of the
cyclic group scheme of order N over A, N, into E. Let M(A, k, N) denote
the space of classical holomorphic modular forms of weight k and level N
over a p-adic ring A. This A-module can be viewed as containing functions
f of isomorphism classes of triples (E, w, L), consisting of an elliptic curve E
over A, a nowhere vanishing differential w, and a level N structure . These
functions are natural with respect to A and are homogeneous of degree -k
in w, meaning that for any A E A, f(E,Aw,t) = A-kf(E,w,t).
The qexpansion of a modular form f is defined to be the value of f on the triple
(Tate(q), Wcan,Ltcan). Tate(q) is the elliptic curve with canonical Weierstrass
equation
where h 4 =-5S
y2+xy=x 3 +h 4x +h 6
3
and h 6
= -5s+7S
12
5
5Sfor Sk
differential is dx/(2y + x) and Lcan() =
full details.
If p
=h
=
"mod q".
~~nk
The
canonical
We refer to [Kat76] for
n
cnonical-q
5 then the Tate curve can be given by the equation
y 2 = 4X 3
E 4 + E6 := 4X392X
12
216
33
+ 93
with canonical differential dx/y. In this case M(A, k, N) consists of the homogeneous polynomials of degree k in the Eisenstein series E4 and E 6 (which
have degrees 4 and 6 respectively) with coefficients in A (or equivalently polynomials in 92 and g93).The Eisenstein series have q-expansions
00
E 4 (q) = 1 + 240 E.o
3(n)qn
n=1
and
oo
as(n)qn
E6 (q) = 1- 504 o5E
n=1
where ak(n) is the sum of the kth powers of the divisors of n, which gives an
explicit way to compute q-expansions in general.
Now let us suppose that B is a p-adically complete discrete valuation ring,
and that K is its field of fractions. We remark that this means that B is the
topological inverse limit of the discrete rings B/pnB, and that the localization,
Bp, of B at its maximal ideal can be viewed as the ring of K-endomorphisms
of K/Bp. More pertinently though is that this condition assures us that any
elliptic curve over B, with ordinary fibres, admits a trivialization. The module
of divided congruences of weight less than or equal to k is defined by
k
Dk(B,N) = {f
e
M(K,j,N)If(q) E B[[q]]}.
j=0
The ring D(B, N) of divided congruences of modular forms defined over B is
the direct limit over k of the modules Dk(B, N). The point of this is that the
individual summands in f need not have q-expansions with coefficients in B,
but that the denominators are cancelled in the whole sum.
Katz showed that the ring of divided congruences (or rather its p-adic completion) is, in some sense, the coordinate ring of the moduli problem of 'elliptic
curves over p-adic rings together with isomorphisms of their formal groups
with the multiplicative group Gi'. More precisely, for a p-adic ring A we consider the set of isomorphism classes of triples (E, A, ) where E is an elliptic
curve over A, 0b: E
Gm is an isomorphism of formal groups over A, and
L
is a level N structure which composes with 0 to give the canonical inclusion.
This set has the structure of an algebraic stack M. The functor which assigns
this set to a p-adic ring A is representable by a p-adic ring W = W(Zp, N)
(which should be thought of as the coordinate ring of the stack). This means
that for any p-adic ring A, the set of (p-adically) continuous homomorphisms
from W to A (or equivalently the stack morphisms from SpecA to M) can
be naturally identified with M. The elements of W are called generalized padic modular functions. Restricting attention to B-algebras, for a p-adic ring
34
B, we obtain in the same way the representing space W(B, N) = W6B of
generalized p-adic modular functions over B.
Given a triple (E, b, ) and a generalized p-adic modular function f we can
apply the homomorphism corresponding to (E, 0, t) to f to obtain a value
f(E, q, L) E A. (Note that we regard ourseves as evaluating f on the triple.)
The q-expansion f(q) of f is defined in this setting by evaluating f on the triple
consisting of the Tate curve over B((q)) with its canonical trivialisation and
level N structure. The key facts are that the q-expansion map W(B, N)
B((q)) is injective with flat (over B) cokernel, and that the smallest ring that
f is defined over is that generated by the coefficients of f (q). These facts are
called the q-expansion principle. The function f is called holomorphic if its
q-expansion lies in B[[q]]. The ring of holomorphic generalized p-adic modular
functions over B is denoted V = V(B, N). Clearly the q-expansion principle
remains true if we replace W by V.
We can regard classical modular forms f as generalized p-adic modular func-
tions by identifying f with the function whose value on the triple (E, 0, ) is
given by f(E, O*( d't) , l), where dt/(1 + t) is the canonical differentialon Gm.
With this identification, Katz proves the following theorem:
Theorem 4.2 If B is a p-adically complete discrete valuation ring then there
is an injection D(B, N)
V(B, N). Moreover the image is dense.
Proof
The map is constructed using the q-expansion principle. Given I E D(B, N)
the fact that B is a DVR implies that (q) is torsion in B((q))/V and so by
flatness, there is some f E V with the same q-expansion as f. The proof that
the image is dense consists of explicitly constructing certain divided congruences d which form a sequence of Artin-Schreier generators for V. We refer
to [Gou88] for details.
El
The second statement in this theorem refers to the p-adic topology. It follows
that the p-adic completion lim D(B, N)/pnD(B, N) of D(B, N) is equal to V.
It remains to explain in what sense V is the coordinate ring of a moduli
problem. V = V(Zp, N) can be constructed as a limit of rings Vn,m, which in
each case are the coordinate ring of an affine scheme. For M > 0 this scheme
is M(Np
m ) ®Z/pn, where M(Np m
) is obtained by adding cusps to the moduli
space of elliptic curves with an arithmetic level Npm structure. For m = 0 we
have to delete the supersingular points in M(N) to obtain the affine scheme
of ordinary points, and Vn,0 is the coordinate ring of M°rd(N) 0 Z/pP. Then
V = lim lim Vn,m.
n
m
35
As above we take the completed tensor product over Zp with B to obtain
V(B, N).
An alternative approach, which we shall use below, is that described in [Kat75]
The ring VB is defined to be the representing object for the functor MtriV(A)
of isomorphism classes of trivialized elliptic curves over a B-algebra A. Mt iv
is constructed from the stack of trivialized elliptic curves by passing to isomorphism classes. The q-expansion map is defined by evaluation on the Tate
curve with its canonical trivialization and using this to define D = DB, the
q-expansion principle and theorem 4.2 still hold. This approach is preferable
because in the above we need N > 3 for M°rd(N) to be representable by a
scheme. To recover VB from V(B, N) we need to take the ring of invariants
under the action of GL(2, Z/N).
4.3
Formulae for the generators of D
The generators dn for the ring of divided congruences constructed by Katz in
his proof of theorem 4.2 are not uniquely determined. For p > 5 Katz gives
a formula for a particular choice of dn in terms of the canonical differential.
Specifically if we set t = x/y and expand w = dx/y in terms of t we obtain
w = E ant'- 1dt
n>1l
for some an E Z[g2 , 93].The dn can be described by saying that an is the n t h
Witt polynomial in the d/. This means that
n
apn=
i=O
Given ap. for all n we can construct dn inductively from this formula by rearranging and dividing by pn. In appendix B we carry out the computation of
an for n < 125, expanding and correcting the formulae given by Katz. This
gives us the following explicit formulae for certain dn.
p= 5:
The coefficients of t 4 , t 24 and t' 2 4 are 16g2,1081344(240gg2 - 7g6 - 60g) and
995569969321779190762968186882(14726735444678400g38g2
_ 218
3
8
+2908841831540736Og 2gg2 - 10570523441402496093gOg95
+43119208g3 ° - 609616768696320093 g2
+888298720100352009g69g - 288697084032614400g 3 4g92
+37563035811840g2 0 + 14430842538840g4g24
-780963243278400g36g2 1 - 72682814985gg9227 )
36
respectively. Therefore we have
do = al = -2
169g2+ 32
a5 - d
5
a 25
d2
1081344(240g3g2
7569408
-
6
'25_5 92+
-
1048576 5
15625 92
16777216 2
3125
5
5d5 -do
-
25
(1692+32)5 + 1048576
7g926- 60934)
25
51904512
3
2097152
-
3125
2
9293g5
g3
5
4
93
8388608 3
3125 92
20937965568
4
92
16777216
3125
12976128
+
15625
and d3 can be written down from this, but is too cumbersome to print. Note
that in Katz original paper a = -2 was erroneously replaced by 1 in the
formula for d. Given these formulae and the q-expansions for 92 and 9g we
can check that q-expansions of these forms do indeed lie in [1/6][q]. For
92-
92
example 1692+32
has q-expansion
5
1 16( +
00
§(T (1 + 240E u3(n)q ) + 32)
n=1
400
1
=5(-
20 +
+ 320 a3(n)qn) -
3
n=1-
oo
64E 3(n)qn.
n=1
p= 7:
From our calculations in appendix B we have
a7
9693
=
a4 9 = 997103501312(6336g8 + 236544g23g6 - 221760g4g26+ 18480g2g9
and so for p = 7 we find
do
-2
96
128
dly 93+-7
2
6317647784312832 8 33694121516335104
7
9
49
93 +
37
-
91g12)
31588238921564160
4 6
26323532434636802_
12962345517056
9392 +
7
Z929
7
'
75144747810816
5764801
93
890604418498560
4
823543
7
100192997081088
823543
823543
3
400771988324352 5
93
823543
6
3
1187472557998080
12
3
949978046398464
3
823543
2
'
As before we can check that these forms are defined over Z[1/6]. For example
the q-expansion of d is
o5 (n)q n + 128)
1 -96 (1 - 504
n=1
1 1148
- ( 9 +224 E Us(n)qn)
n=1
164
=
+ 32
a(n)qn.
n=1
p-ll:
We give one more calculation, for p = 11. Again from appendix B we have
all = -25609293
a 121 =
°
-272699558638918420400406398026907648g92
4g
92
+721965202905751390557818724689134498086912000g3
1
+38443459611963324456100955940478173315072009g3g
-159592308010745044228570454931282362735001600g3
692
-64449329349467926294051602606095761145856000g8 8
+400690687612692021759589392202469646438236160g
3Og25
+6418996802845451753788655804174252900352000g3
8
-78349442143946992777379665639561494528000gg
42
2
-92077910537260035303339986591981661176463360093g2
-10135258109755976453350509164485662474240g32
0
+428578145060906627158316119737772277760g3g22
7.
Therefore
do= -2
2560
di
-11
9392 +
38
2048
g23
349485996472393858691826872186165211955200
d2-=a92
11
583545163895041068526241436743113900032000
+
21
18
11
93g2
65633200264159217323438065880830408916992000
14
9
93 92
11
+
3
920779105372600353033399865919816611764633600
121
12
12
9 3 92
921387100886906950304591742225969315840 20
93
11
4 24
7122676558540635707034515058141954048000
11
+
g392
15
1o
36426426146608365614508126563860876948930560
9211
13
5859029940860720572186509327826887376896000
8
g32
11
38961649550991511559846919976161116160
+
11
2
27
9392
14508391637340458566233677721025669339545600 16
11
9392
24790868967174401854582399820627968
11
6
30
92
68953501833760746541571357357938184604037939200
+ ~3138428376721
30948500982134506872478105600000000000
+
3138428376721
24758800785707605497982484480000000000
285311670611
99035203142830421991929937920000000000
±+
285311670611
11
3
2
10
10
93 92
9
9
9392
237684487542793012780631851008000000000 8
8
9392
285311670611
7 7
380295180068468820449010961612800000000
+
285311670611
9392
425930601676685078902892277006336000000 6
285311670611
6
32
340744481341348063122313821605068800000 5 5
+
285311670611
g9392
4 4
194711132195056036069893612345753600000
9392
285311670611
39
77884452878022414427957444938301440000 3 3
+
285311670611
9293
20769187434139310514121985316880384000
2
285311670611
g3 g2
2
3323069989462289682259517650700861440
+
285311670611
Let us try to show that the q-expansion of-29°g39
93922
+ 2048
has coefficientsin
Z[1/6]. If we multiply by 11 then the constant term is 502 + 2048 = 6__ .
Since 165968 = 11 x 15088 this coefficient is okay. For n > 1 the coefficient of
qn qis
-504,55(n)
l o-1
a(k)o5(n-k)) so it is necessary
is so (240,3
(240 3(n)
(n)-504
(n) -240
- 504
504
to show that the term inside the parentheses is divisible by 11. Reducing mod
11, then dividing by 2 this becomes 5 (n) - o3 (n) -2 Ek=l 3 (k) 5 (n- k)
which should be divisible by 11. A direct proof of this congruence has proved
to be elusive, which demonstrates how remarkable it is that every coefficient
in these q-expansions lies in Z[1/6]. In the course of searching for a proof we
did however uncover the following fact.
Proposition 4.2 Half the number of integer solutions to the equation
x2 +xy +3y 2 = n
is equal to o.5 (n) mod 11.
Proof
The number of
The form x 2 + xy + 3y2 has discriminant 1 - 4 3 =-11.
equivalence classes of quadratic forms of discriminant -11 is equal to the
number of triples (a, b, c) of integers satisfying b2 - 4ac = -11 and either
-a < b
a < c or 0 b a = c. (See [Ken82] p323) In this case we see
that b2 - 4ac < b2 - 4b2 = -3b 2 so we must have b = ±1. In either case the
0
inequalities force a = 1 and c = 3. However for b = -1 we contradict b
and b > -1 so we must have b = 1. It follows that x2 + xy + 3y2 represents
the unique class of forms of discriminant -11. A theorem due to Dirichlet
([Ken82] p307) states in this case that for n > 0 prime to -11, the number
s(n) of solutions to x2 + xy + 3y2 = n is given by
s(n) = 2E ( n
din
where (L) is Kronecker's symbol. Specifically (ll) is zero if 11 divides n, 1
if n is a quadratic residue mod 11 and -1 if n is a quadratic non-residue mod
11. Both this expression and a5(n) are multiplicative so we can assume that n
is a prime power. If n = pr where p is not 11 then cr5(n)= ~ = 0 p5g. If p8 is a
40
quadratic residue mod 11 then p5 8 is a tenth power so is 1 mod 11. Otherwise
p 5 8 = -1 mod 11. It follows that o 5 (n) mod 11 = s(n)/2 in this case.
If n = 11r we claim that there are exactly 2 solutions to x2 + xy + 3y2 = n
and so the formula holds in this case too. For n = 11 it is easy to see that the
only solutions (x, y) are (-1,2) and (1,-2) so this is okay. For higher powers
of 11 we will show that both x and y are divisible by 11 and so by dividing by
121 and using induction on r we will be done. (We should note that the only
solutions with n = 1 are (1,0)and (-1,0).)
To prove that x and y are both
divisible by 11 we use the standard algebraic reduction that says for a solution
(x, y) to x 2 + xy + 3y2 = n we obtain a solution (X, Y) to X 2 + 11Y2 = 44n
where X = -11ily and Y = 2x + y. If 11 divides n then 121 divides 11Y2 and
so 11 divides Y. Now for r > 1 we have 113 divides X2 and so 11 divides y.
This and the fact that 11 divides 2x + y allow us to conclude that 11 divides
x.
[5
The same proof will show that 03(n) mod 7 is half the number of integer
solutions to x2 + xy + 2y2 - n. Indeed whenever we have a prime p of the
form 4k + 3 (k > 0) for which x2 + xy + (k + 1)y2 represents the unique class of
forms of discriminant -p, we will find that 2k+1 (n) mod p is half the number
of solutions to x2 + xy + (k + 1)y2 = n. Note that p = 3 doesn't work because
the automorphism group of x2 + xy + y 2 has 6 elements rather than 2. The
next primes that work are p = 19, p = 43 and p = 67. A necessary condition
for x2 +xy+ (k + 1)y2 to represent the unique class of forms of discriminant -p
for larger primes is that both k + 1 and k + 3 must be prime, for if either were
composite, the factors would give different values for a and c that represent a
different class of forms of discriminant -p. The only part of the proof which
is not clearly the same is that (-1,2) and (1,-2) are the only solutions for
n = p. To see this we change the equation to (2x + y) 2 + py 2 = 4p. It follows
that IyIis at most 2. The values y = 1, 0,-1 are easily ruled out if p > 3 since
neither 3p nor 4p is a square. This just leaves us with the two stated solutions.
Higher Primes
The formulae for ap in appendix B for larger primes p will give many more
congruences mod p between 3 (m) and o 5 (n) for various m and n. These
formulae will be quite large for any primes bigger than 11, and so are perhaps most useful simply stated as congruences for the q-expansions of various
polynomials in g92and 93.
4.4
D as Kotmf.
In section 4.2 we defined the ring of divided congruences and explained how
its p-adic completion can be viewed as the coordinate ring of a certain mod41
uli problem. Next we show that this ring is precisely Kotmf. We give two
related but philosophically different approaches to this calculation, based on
two different characterisations of tmf. The first constructs tmf directly as a
homotopy inverse limit and then uses various homotopy theory results to calculate Kotmf as an inverse limit. The second approach is to describe K Atmf
directly as a stack of trivialized elliptic curves, as considered above, and then
to prove that this stack is in fact a scheme. This approach is more direct, and
in fact will be used to complete the calculation from the first approach, but
the first approach has the advantage of being described using only homotopy
theory.
The following lemma is the key algebraic step.
Lemma 4.1 If J is an elliptic curve and T + O(T5 ) is a coordinate near
infinity. Then there exists a unique pair (x', y') of Weierstrass coordinatesfor
J such that x'/y'
T mod T5 .
Proof
If we choose an arbitrary pair (x, y) of Weierstrass coordinates then any such
coordinates (x', y') are related to (x, y) by a transformation of the form x' =
A-2x + r, y' = A-3 y + A- 2 sx + t. We can expand x and y as Laurent series in
T to get x = T-2(Uo+ ulT + u 2T 2 +" ) and y = T-3 (vo+ vT + v2T2 +.. )
where u0 and vo are invertible and ug = v2. Substituting these power series
into x' and y' and expanding x'/y' as a power series in T (with the help of
Maple) gives
2)
x/y = luoT _l(-uiVo + UoV
1 + lsuo T2
--
A(-uovo2 -2uovoAsu
1
V2
Vo
V3
+U2V?2 +rvO2A2 - Ulv OV1 +uOVl2 + 2U2V AS+ A2S2UO3)
Vo
v
+
3 t- u o3 + UlV2V + Ul2v o 2
As- 2vouovv 2
-A(2uov~2su2A + UoV2V3 + UoVO2A
2
3
-4V
o
UOVlAsu1 - 2V0oAsu2
-ulvov
2 2
s uUi
- 3voA
0
2
3UVA
+ uov3 +
+ U2V2Vl + rvOA 2vVl + rv2 A3 suO
+ 3U3V1 A 2 s2
+ A3 s3 u4)T 4 /v 4
+ O(T 5 )
We have to solve for A, r, s and t such that this expression is T mod T 5 . This
gives us 4 equations in 4 variables (from the coefficients of T, T 2 , T 3 , T4 ) with
Jacobian matrix
VO
0
0
0
2
"%-? *
~0~~~~~)
0
A
Va
0
0
42
The first equation gives A = U0 and so the Jacobian is invertible. The inverse function theorem therefore tells us that there is a unique solution to the
equations. In fact with some more algebra we find that this solution is
V0
Uo
S =
r
VI
U1
Vo
U0
-
-'
(-U 3 UoVo + V2 U
-
t =3
(-U3UoVo + V3 uO -U
+ U2Vo)Uo
1VUllUO
2
VUO + U2 UlVo)Uo
The spectrum tmf is constructed to have the mapping properties of a universal
elliptic cohomologytheory. Specifically,we can construct a functorial diagram
of Landweber exact elliptic cohomology theories, El, each of which is an Eoo
spectrum satisfying further properties from obstruction theory. This means in
particular that each Ei is complex orientable, and the formal group associated
to this orientation comes with an isomorphism to the formal completion of
an elliptic curve E over Ei*. Furthermore on finite CW-complexes X we have
Ei*(X) = E* ®Mmu*MU*(X), where the MU* action on E* comes from the
complex orientation. For any spectrum R, there is a map from R A tmf to
the homotopy limit Holim(R A Ei). The first result from homotopy theory is
that this map is a homotopy equivalence. A proof of this statement would
involve the nilpotence theorems of [Smi88] but it is certainly reasonable given
the relation of Ei to MU.
This equivalence means that for any R, there is a spectral sequence
lim
rt(R A Ei)
==>
rt- 8(R A tmf).
The second result that will allow us to calculate 7r*(KA tmf) = K*tmf is the
following.
Proposition 4.3 If R is complex orientable then the derived functors lim2
are zero for s > O. In particular,
7r*(RA tmf) = lim REi.
Proof
We may assume that R is even and periodic, so that we can associate to
it a formal group GR. This is sufficient because it is true for the periodic
43
version of MU, which we denote MU{u'l}, and we can extend to other R by
naturality. In this case R A tmf can be identified with the stack of elliptic
curves E together with an isomorphism of E with CR. (For R = K this would
be the stack of trivialized elliptic curves.) If J is the universal trivialized
elliptic curve over W then this stack is naturally identified with the stack
of isomorphisms Iso(J, GR) (since such isomorphisms classify such E). We
can think of the R A Ei as forming an etale cover of this stack. Then by
definition, the 5 th cohomology group of the stack is the lim' that we want.
A basic fact about the category of stacks is that it contains the category of
schemes as a subcategory, and that the sth cohomoloy of a scheme is zero
for s > 0. Therefore it remains to show that Iso(J, GR) is a scheme. If we
choose a coordinate on GR then pulling back that coordinate defines a map
from Iso(J, R) to the stack A of coordinates on JR. A can be identified
with MU{u± 1} Atmf. We claim that A is an affine scheme and so this map
identifies Iso(J, GR) as a closed subscheme. This follows from the lemma: If
we let bn denote the coefficients of T' for n >)5 then
A = SpecZ[al, a 2 , a3 , a4 , a6][u1][b 5 , b6 , ...].
The details of this can be found in [Rez]. Note that an element of r2n is
D
identified with an element of ir0 by multiplying by uE.
Corollary 4.1 For any p-adic ring B, the ring Ko(tmf; B) can be identified
with the p-adic completion of ring of divided congruences DB, that is VB.
Proof
The previous proposition gives us two different ways to see this statement. The
direct approach is to observe that Spec iro(K Atmf) = Iso(Js, Gm) = Spec VB
Under this identification, forms of weight 2i must be multiplied by v -1 (v being
the Bott class) to put them in 7r0. The alternative method is to calculate KEi
directly as in [Lau99]and use the statement of the proposition to obtain Ktmf
o
as a limit.
Corollary 4.2 There is an isomorphism of Zp-modules
K° (tmf ; Zp) - lim hom (Vz/pi,2/p
i)
for any prime p.
Proof
This follows from proposition 4.1 and the previous corollary. One simply needs
D
to note that the continuous character must map pi-torsion to pi-torsion.
In section 5 we will expand upon this description and then apply Bousfield's
theorem to obtain K* (Qtmf;
Zp).
44
4.5 Diamond operators
The ring of divided congruences has a natural action of Zp by what are called
diamond operators. The p-adic K-homology of a spectrum also has an action
of Zp given by the stable Adams operations. In this section we define each of
these actions and show that they correspond under the identifications of the
previous sections.
The diamond operators act on generalized p-adic modular functions through
varying the trivialization. Specifically, given a generalized p-adic modular
function f and a E Zp we define (a)f by the equation
1 0).
(a)f (E, ) = f(E orThis action preserves the ring V and so gives rise to an action on D.
(a)f =
akf
If
then f is said to have weight k. Under the identification of
classical modular forms as generalized p-adic modular functions, a form of
weight k becomes a function of weight k. It follows that if f = E i
D
wherefi has weighti then (a)f = E aifi.
The stable Adams operations are natural transformations of K-homology. For
an integer k they can be defined axiomatically as follows. From the work of
Miller [Mil89] it follows that natural transformations of a Landweber exact
homology theory are determined by their action on coefficients and on line
bundles. As we noted in section 3, the operation ok acts by sending the Bott
class v to kv, while it acts on line bundles by raising them to the kth power.
We remark that Miller phrased this in the language of formal group laws.
In this language ok is described as the ring endomorphism of K, = Z[v ± 1]
determined by sending v to kv, together with the strict isomorphism
[k]: Gm ?ikm
of the multiplicative formal group law. To see this we just have to note that the
multiplicative formal group law x + y- xy corresponds to the tensor product
of the line bundles (1 - x) and (1 - y). Of course this definition requires us
to be able to divide by k so over the integers we only obtain two operations.
Working with p-adic K-theory we obtain Obk for any k prime to p. We can
then extend this to any p-adic unit by continuity.
Given this description of the stable Adams operations, we can easily prove the
result that we want.
Proposition 4.4 Under the identification of Kotmf with the completion of
D, the Adams operation ok A 1 correspondsto the diamond operator (k-i).
45
Proof
We showed above that if f = E fi E D where fi has weight i then ()f =
E aifi. Thus (k-l)fi = k-fi. Our identification of Kotmf with D requires
us to multiply fi by v-i to put it in degree zero. Therefore we have (k A
1)(v-ifi) = (k(v-i)f i = k-=v- fi, and so opk A 1 also acts by multiplying fi
by k- .
0,
In terms of the representing spectra, if we think of K A tmf as the stack of
trivializations of the universal elliptic curve, then the diamond operator (k-1),
which act by multiplying the trivialization by k is represented by a map of the
spectrum K. This map is precisely the Adams operation obk.
5
5.1
The K-cohomology of tmf and Q tmf
Hecke Algebras and duality
In section 4 we showed that there is an isomorphism of Zp-modules
K*(tmf; Zp) - lim hom (Vz/pi,Z/p i)
for any prime p. There is a duality between the ring of generalized p-adic
modular functions and the ring of Hecke operators acting on V. Given the
formula above we can expect there to be a relation between the K-cohomology
of tmf and this Hecke algebra. We will use this relation below to obtain a
description of the K-cohomology of Qfttmf.
For the moment let us assume that p > 5 and that is an integer prime to p.
We recall the definition of Hecke algebras from [Gou88] The Hecke operator
T acts on the ring V of generalized p-adic modular functions as follows. If
(E, k)is a trivialized elliptic curve then for every subgroup H of order 1, the
projection r: E -->E/H induces an isomorphism on formal groups. Therefore
we obtain a trivialized elliptic curve (E/H, T7r'). If f E V then T}f is defined
by the formula
Tf (E, q) =
E f(E/H,
7r-1)
where the sum is over all subgroups of order 1. If f has q-expansion f(q) =
anq" then
Tf (q)=
alq + (()f )(ql).
For = p it can be shown [Gou88] that there is a linear operator Up on V
which on q-expansions sends E anqn to E anpq . If we denote Up by Tp then
we obtain T1 for all according to the following formulae:
46
1. T is the identity.
2. If and m are coprime then Tlm = TITm.
3. If is a prime distinct from p and k > 2 then
Tlk=
TITk-l -
()Tk-2
4. For any k > 0, Tpk= Tk
It follows from these formulae that any Tl can be written as a polynomial in
the Tq for q a prime dividing 1. Furthermore the Tq commute with each other.
It is clear from the modular definitions that the Hecke operators commute
with the diamond operators as well.
Definition 5.1 If V' C V is a subring that is preserved by the Hecke and
diamond operators then the Hecke algebra Tv, of V' is defined to be the com-
pletion of the commutative subalgebraof the space of Zp-linear endomorphisms
of V' generated by the Tl for 1 prime to p, Up and the diamond operators. The
restricted Hecke algebraTv, is obtained by omitting the Up operator.
As two particular cases we have T = Tv and TD', where D' is the ring of
divided congruences of forms of positive weight. To see that D' is preserved
by the Hecke operators we can look at the effect on q-expansions. The only
concern would be the term j, but the effect of the diamond operator (1) on
a form of weight i is to multiply by l i , so if we restrict to forms of positive
weight this will cancel the 1 in the denominator.
Now let us consider the ring VB for a p-adic ring B. For any V' as above there
is a bilinear form
Tv,
V'
B
(T,f) , ) a(Tf)
where a (Tf) is the coefficient of q in the q-expansion of Tf. In particular
if T is the Hecke operator T and f has q-expansion f(q) =
aqn then
(Tl, f) = al. Varying we should be able to determine all of the coefficients
(except the constant term) in the q-expansion of f and hence determine f
itself up to an additive constant. This means, for example, that if we restrict
attention to space VB/arof forms with zero constant term (parabolic forms)
then we obtain an injection
Vpar
horn (Tvpar B)
47
where the right hand side consists of continuous (with respect to the p-adic
topology) homomorphisms of B-modules.
There are several results which strengthen this observation; we mention two.
Proposition 5.1 Let B be a p-adic ring. Then the map f -, (, f) induces
an isomorphism of Zp-modules
Var-
hom (Tvz ar , B)
(continuous Zp-module homomorphisms). Moreoverf correspondsto an algebra homomorphismif and only if it is an eigenformfor the Hecke and diamond
operatorsand a (f) = 1.
Proof
We refer to [Gou88] for details. The condition al (f) = 1 corresponds to the
condition that ring homomorphisms must respect multiplicative units. Notice
that the ring B occurs in the right hand side only as the target space, and not
o
as part of the Hecke algebra.
Proposition 5.2 Let V be a p-adic ring and defineVB = (VB+Qp)/Qp. Then
the map f -. (., f) induces an isomorphism of Zp-modules
VB - hom (Tvz, B)
(continuous Zp-module homomorphisms). Moreoverf correspondsto an algebra homomorphismif and only if it is an eigenforrnfor the Hecke and diamond
operators and al (f) = 1. For B = Zp there is also an isomorphism
Tvz,
hom (Vd,, Zp)
induced by the same pairing.
Proof
See [Gou88]. We note that VB has a dense subset consisting of sums of forms,
o
whose q-expansion lies in Qp + qB[[q]].
We are interested in the inverse limit over i of the spaces hom (Vzp, Z/pi).
We remark that these spaces consist of module homomorphisms. If we consider the subspace of algebra homomorphisms then this space would be exactly
the set of isomorphism classes of trivialized elliptic curves over Z/p i , by definition of V. Also we note that this inverse limit does make sense, since VB is
contravariant as a functor of Zp-algebras B.
48
Now let p be any prime and let Pi: Z - Z/p i denote the reduction mod pi
map. The map induced by Pi on spaces of generalized modular functions is
denoted pi*. For each i there is a map
hom (Vzp, Zp) (i)
hom (Vzp, Z/p
i)
(_* hom (Vz/pi, Z/p
i)
which give rise to a map
p: horn (Vz,, Zp)
lim hom (Vz/pt, Z/p').
The final statement in proposition 5.2 also shows that the inclusion of Vz in
Vp induces a map
q: Tvzp - hom (Vzp,Zp)
when p
5.
Proposition 5.3 The map p: hom (Vzp,7Zp)
-+ lim._hom (Vz/pi,Z/p i ) is an
isomorphism, while q : Tvz - hom (Vzp,Zp) is injective but not surjective.
Proof
The map 0 is not surjective as any map from Vzpto Zp which does not vanish
on the constants does not extend to V1p.It is injective because a map from Vp
to Zp must vanish on the constants so is determined by its restriction to Vzp.
To prove that p is an isomorphism we first note that Vzpis the direct limit of
the Vz1/p, as an algebra, but we can regard the same direct system as a diagram
of Zp-modules and obtain the same object as a direct limit of modules. Given
this the statement follows by some definition chasing. Specifically an element
of the direct limit consists of a family of maps from Vz/pito Z/p i . These maps
factor through each Z/pk for k > i and so factor through Zp. But such a
family of maps is precisely a map from the direct limit of the Vz/pi to Zp. El
Corollary 5.1 There is an isomorphism
K*(tmf, Zp) - - hom (Vz, Zp)
for all primes p and a split inclusion
Tvzp * K*(tmf, Zp)
for everyprimep
5.
Proof
This is mostly just a restatement of proposition 5.3. To obtain a splitting of
b we note that elements of Tvz can be viewed as elements of hom (Vzp,Zp)
which vanish on the constants. Such an element can never be a multiple of
a map which doesn't vanish on constants, and so we can construct a right
inverse to 0 by extending the identity map to zero on functions which do not
vanish on constants.
El
49
5.2
K*(Q tmf; Zp) for p > 2
From corollary 5.1, our calculation of the first and second cohomology groups
of tmf and Bousfield's theorem, we have the following statement.
Theorem 5.1 For p > 3 there is an isomorphism
K*(D tmf;Zp) 2 T(hom (Vzp, Zp))
whilefor p = 2
K*(Q tmf; Z2 ) - W(hom (V 2 , Z2 )
Z/2).
Let us recall the definitions of T and W to make this statement explicit. T is
left adjoint to the functor sending a connective Z/2-graded p-adic OP-ringR
to the Z/2-graded p-profinite abelian group R. Therefore for p > 2 and any
connective Z/2-graded p-adic OP-ringR, there is an isomorphism
homop (K*(Q tmf ; Zp), R)
homAb (hom (Vzp,Zp), f).
Similarly W is left adjoint to the functor sending a Z/2-graded 2-adic 02 -ring
R to the map A -+ R/ 2 R.k Therefore for any R there is an isomorphism
home2(K*(f2'tmf; Z2), R) - homAb(hom (V 2, Z 2 ) - Z/2,
- R//Fr2).
Let us consider a particular example for p > 2. If Y is a connected CWcomplex with Hi(Y; Zp) = 0 for i < 3 then X = K*(Y; Zp) is a connective
Z/2-graded p-adic P-ring. In particular Y - S2n for n > 1 satisfies these
hypotheses and according to [HatO1]K* (S2n) = Z generated by (H - 1) *... *
(H- 1). It follows that the reduced p-adic K-theory of S2n is a single copy
of the p-adics and there is an isomorphism
hom 0 p (K*(Qo'tmf
; Zp), K*(S 2 n; Zp))
homAb (hom (Vzp, Zp), Zp).
There is a natural map
K*
.r2nro tmf
homep(K*(Qotmf; Zp), K*(S2n; Z))
given by applying K-theory to a representing map from S2n to QcOtmf. There
is also a map
e: Vz -
sending v
E
homAb (hom (Vzp, Zp), Zp)
Vzpto the map which sends a map to its value at v.
Lemma 5.1 The evaluation map e is injective but not surjective.
50
Proof
If e(v) = 0 then f(v) = 0 for all f E hom(Vzp,Zp). However if v ~ 0 then
since Vz, is torsion free, there is a map sending av to a and the rest of Vzp
to zero. Thus if e(v) = 0 then v = 0 and e is injective. It is not surjective
because the space of group homomorphisms is much larger than the space of
Zp-module homomorphisms. For any copy of Zp in Vzp we obtain a copy of
the completed group ring Zp[[Zp]] in the space of group homomorphisms.
We remark that the copies of Zp[[Zp]] correspond to the action of the Adams
operations bk for k prime to p. The P operation on the other hand is introduced freely on the left hand side.
A natural question to ask is whether any elements of the homotopy groups
of fQOtmf correspond to elements of Vzp. For p > 5 we have the following
theorem.
Theorem 5.2 With 6 inverted, the homotopy of Qtmf
is
ir*Q0tmf - Z[, E4, E6].
Proof
For details of this computation and some history see [Bau].
D
Corollary 5.2 For p > 5, the elements pidi E Vzp correspondto elements of
the homotopy groups of QOtmf.
Proof
From our formula for d/ in terms of ai in section 4.3 we know that pidi is a
polynomial in 2 and 93 and so this result follows from the theorem and the
0
equations 1292= E4 , 21693 = E6 .
The map 0: Tvzp -+ hom (Vzp,Zp) from corollary 5.1 gives us another way to
look at K*(Qotmf; Zp). Applying the functor T we obtain a map
Tq: T(Tvzp) - K*(Qotmf ; Zp).
The Hecke algebra Tvzp is a module over the ring Zp[[Zpx]]under the action
of the Diamond operators. It follows that each Hecke operator T1 generates a
free A-ring on one generator when we introduce OPfreely with the functor T,
and this gives all of T(Tvzp).
Let us summarise:
51
Theorem 5.3 For p > 3, K*(Qotmf; Zp) is the free 0P ring generated by
the group hom (Vzp,Zp).
The A-ring structure is given by the Adams op-
erations, which correspondto the diamond operators on V. If p > 3 then
K*(Qotmf;
Zp) contains the free 6P ring generated by the Hecke algebra Tvz,
as a direct summand.
Proof
The only part which requires comment is the last statement. The map :
TvzP - hom(Vzp,Zp) is a split inclusion of p-profinite abelian groups. It
follows that To is a split injection also.
0
5.3
K*(Qotmf;
2)
For p = 2 we found that there is an isomorphism
K*(Q' tmf; Z2) -W(hom
(VZ2,Z2)
/2).
We have some understanding of the functor W from our examples in section
2.4. We know that
W(Z 2 - Z/2) - Z2 [[x, 02x, ...1]],
and
W(zP[[zp ]] - zA
zp[[X, 2(X), ...]]
for all p. We are interested in Vz2 as a Z 2 [[Z ]]-module under the diamond
operators and so we would like to understand what we get if we apply W to
the composition
Z[[z2x]]-z2- z/2.
More generally we have the following construction.
Lemma 5.2 For any compositionM -- H there is a composition
K of p-profinite abeliangroups
W(M -- H) -- W(M --+K)
W(H --+K).
Proof
For any R there is an isomorphism between homp W(M -* H); R) and the
set of commutative squares
M H
R
R/r2?
-
52
and similarly for the other maps. Given such a square for M -- K we obtain
a square for M - H since the left hand arrow factors through H. Given a
square for H
K we obtain a square for M - H by mapping M onto the H
in the square. This means that we have a composition
hom0o (W(H -K);
R)
homop(W(M -+ K); R) -4homop (W(M
H); R)
for any R. Now if we take R to be W(H
K) we obtain a map W(M
H) - W(M -- K) as the image of the identity of W(H
K). Similarly
we obtain a map W(M
K) -- W(H -+ K) as the image of the identity of
W(M
K).
[]
We could also view the maps in the lemma as the result of applying W to the
squares
~I
and
'I
$
which are maps in the category of maps. This allows us to deduce the following
result.
Proposition 5.4 The maps
W(Z2[[Z~']]-- Z2)
-+
W(Z2 [[Z]]
Z/2)
and
W(Z2[[Z
--
Z/2)
W(z 2
-
Z/2)
are surjections. The latter is split.
Proof
The functor W is a left adjoint and so is right exact. This means in particular
that it takes surjections to surjections. The inclusion of Z2 in Z 2 [[Z2]] gives a
square
Z2
z/2
Z 2 [[Z
-
,53
]]
Z/2
and applying W to this square gives the desired splitting map.
[]1
We could replace Z2 [[Z2]] in the above with horn (Vz 2, Z2 ) and reach a similar
conclusion. This allows us to say the following.
Theorem 5.4 The A-ring K*(fQotmf; Z2 ) is a quotient of the free Z/2-graded
p-adic A-ring generated by horn(Vz2, Z2 ). As a P ring is contains the free BP
ring on one generator as a direct summand.
Proof
We only remark that W(Z 2 -- Z/2)
Za[[x,OPx,...]]is the direct summand
coming from the split surjection from proposition 5.4.
El
54
A
Combinatorial proof of the formula for P3,.
In this appendix we give some more details on the calculation of the polynomial
P3,n=--P3 ,n(el, ... , e3n) from the definition of a A-ring. Let us recall the formula
for P 3,.n-
Proposition A.1 Let w = -2 +
i be the cube root of unity. Then for all n
P3,n(el, -e3n)=
cjnlkLelejek
l<j<k,l+j+k=3n
wherec ,s = 2YR(r ) unlessr = O,t, 3t/2. If t > 0 thenfor r = 0,3t/2 the value
of cr,8 is 1 whilefor r = t it is 2R(Wr) + ()t+l.
If t = r = 0 the coefficient
is zero.
The definition of P3 ,n that we use is that it is the coefficient of t 3 in the product lFi1<<...<in
<3n(1 + Xil ' xint), where ei is the ith elementary symmetric
function in the indeterminates xj. Note that the highest exponent of any xi
that can occur in the coefficient of t 3 is 3 so the degree of P3,, is 3. Let us
introduce the notation xr ,, t to represent the monomial
Xrst --= x...
1 '
3
r+l ... XsXr+s+..
r+sr+s+l Xr+s+t
which is a typical monomial in the coefficient of t 3 . In the polynomial P3 ,n there
are two types of monomial that may occur. There are the degree 3 monomials,
which are also exactly those of the type el+lej+lek+l for elejek a monomial that
may occur in P3 ,n- 1 , and there are monomials of the form eke3n-k. Notice that
the coefficient of xr + 'l *' t in el+lej+lek+l is the same as the coefficient of Xr,' ,t
in elejek. In fact in both cases it is equal to p (p)
Z
(P) (t-l+r+p)
where the sum is taken so that all the numbers are non-negative. Here p is the
number of the terms from el that are chosen to lie between r +
and r + s.
Once these are chosen the terms from ej must include the other s-p such
numbers, but then may also have q of the original p as well. We can also see
' l s t is zero in the degree 2 monomials, and that its
that the coefficient of x r+
coefficient in P3 ,n is equal to that of x r s t in P3 ,n-1. Taking these three facts
together we conclude that the coefficient of el+lej+lek+l in P3 ,n is equal to
the coefficient of elejek in P3,n- 1 . This means that we only need to determine
the coefficientsof the monomials eke3n._k in P3 ,.. We remark that if we had
made this same observation when dealing with P2 ,n the calculation would have
become trivial, as we would only have to determine one coefficient.
We have reduced ourselves to dealing with the monomials XOi,3n- 2i. Conve-
niently these are equal in number to the monomials eke3n-k so we expect to
55
obtain an invertible linear system for the unknown coefficients. In fact it will
turn out to be triangular. The full system has the form
B
In this equation
with 1 > 0 and
n- 1 case. The
for > 0 and
xOi,3n - 2i in P3 ,n
0
c
b
c is the vector of known coefficients for the monomials elejek
Bc = b is precisely the set of equations that we have in the
matrices A and L contain the coefficient of xOi,3n- 2i in elejek
= 0 respectively. The vector d contains the coefficients of
and so the system that we have to solve is Lv = d- Ac.
We can give explicit formulae for the entries of d, A, and L. In the formulae
belowwe adopt the conventionthat (n) = 0 if m > n or either term is negative.
These values correspond to counting impossible choices so in our context they
should be zero. This will allow us to have cleaner formulae without affecting
our calculation. For example with this convention the formula
(mn) E±Q)(ia )
holds for any positive values of m and n.
Lemma A.1 If the ith row of d correspondsto xOi,3n-2i then it contains the
number
1 , ri (3n- 2i 2(n -i + 1)8
6 10
if n
n-
, n-i
+ I
i and
1 Ax (i) 3n-2i) 2(n-i + ) 3
6
k, n -i+l 3
ifn=i.
Proof
The coefficient of x0
i
'3n
- 2i
in P3 ,n is the number of ways of partitioning
the
numbers 1, 12, 2, ..., i, i, i+1, i+2, ..., 3n-i into three distinct sets of n distinct
numbers. Let us suppose that we put numbers less than i + 1 in the first set.
from the other
Choosing the first set involves first picking from i, then n(3n-2i)
ways. The second set must contain all
3n - 2i so it can be done in (i)I\n-Z]
the numbers from 1 to i that weren't in the first as we cannot have two equal
numbers in the third. Therefore we just have to pick n- i + numbers from
the set containing the numbers less than i + 1 that are in the first and the
3n- 2i - (n- 1) numbers bigger than i that are not in the first. Note that by
56
our convention we do not need to worry about whether all this is possible. The
product
i) ('nl )! (2i+)) therefore gives the number of orderedpartitions.
product
n-i+l
We don't care about the ordering so we have to divide by 6. The final thing
that we have to worry about is that our three sets are all distinct. This will
be automatic unless two of the sets do not contain any numbers bigger than
i. This can only happen if n = i and either = 0 or = n. If = 0 the second
and third sets are both {1,2,..., i} so this should not be counted. If I = n we
just have to exclude the 2 cases where either 0 or n of 1 through n are in the
second set. This leads us to the expression in the second formula.
l
Next we consider the coefficient of xOi,3n - 2i that occurs in eeiek.
Lemma A.2 If 1 < j < k then the coefficient of xOi3n- 2i that occursin elejek
is
3n-2ia 3n-2i +2a-1
>j i
a=0
'
5£
aJVI-aJV
-i+a
Proof
The argument is the similar to the previous lemma. This time however the
sets do not need to be distinct and the partitions are ordered by 1,j, k so we
neither have to divide by 6 nor rule out any cases.
a
The previous lemma give us the entries of both A and L from our system of
equations. However we should take a closer look at L which has columns that
correspond to the monomials eke3n.-k.
Proposition A.2 Let the (k + )th column of L correspondto the monomial
eke3n-k. Then L is an upper triangular of dimension L2j and in the i row
andjhcolumn
jth column it
it contains
contains (3n-2i).
Furthermore let vo be the vector whose
and
j-i
(k + l)th component is 2R(e
2 rki/3).
If n is even we modify vo by replacing the
2 in the final component by 1. Then every component of Lvo is (-1)n.
Proof
The formula for the entries of L comes from the previous lemma. To prove
the second claim about L we let w = e2 ri / 3 . Consider the ith entry of Lvo. For
n odd it is given by
3n
2Ry.(
3n
-2
k
=
k=i
2
i
LR (E
3n
k=0
= Wi(1 + )3n-2i
= wi(-)3n - 2i
= (-1)3n3n-3i
= (-l)n
57
)k+i
Here the key observation for the second equality is that we sum over exactly
half the terms in the binomial expansion of (1 + w)3n- 2i, but we multiply by
two to account for the other half. The binomial coefficients and the real parts
of the powers of w match up perfectly so that the real parts of the two halves
are equal. It is not true that the imaginary parts are equal though. For n even
the calculation is essentially the same. This time instead of being between
consecutiveoccurrencesof - in the sequence2, - 1, - 1,2, - 1,- 1,2, - 1... we
would have a 2 in the final position. Our modification to v0 splits this 2 evenly
between the two halves and so the second equality still holds.
0
With these results in hand we see that the formula for P3 ,n reduces as in the
n = 2 case to some identities among binomial coefficients. The formula for
Lvo is our guide as to how to proceed with the calculation. We can generalise
it to the case > 0 (where we are looking at monomials elejek).
Lemma
3n-L2
A.3 With w as above we have
(i) (3n-2i) (3n -2i+2a i))2R(wjl)= (-1)
(
j=O a-
In this formula when a 3 ( n- 1 )/ 2
=
2 we replaceit by 1 as before.
Proof
First we assume that
< i and 3n- 2i > 1. We can sketch the region of
summation in the (j, a) plane as shown.
a
i
('An-1V/
\=/~J -
I
i-1
i
j
From this we can see that we can interchange the order of summation and
obtain
f
(
a=O
2- ( 3n-2i +2a-l)2 f-)
3)(3n-2i)
kal-a
- - i +a
:
j=i-a
58
2,w-1
3n-1-2i+2a
a=0
3 nV-i2i
I (z)
(
=E(a)(
1
(3n-2i+ 2a-+
2
k=0
-a
])(1
a
(i)
( )(nl
1 ()3n-2i+2a-l
=Ea=Oa
i\l- (3n-2i)
ia
I:a
a( ) 3n-ia C)32iaa=O
-
a=O
exactly as above. In this proof we use the identity that
(mIn) =±()(Ia)
To see this one just considers how to distribute 1 items among m + n positions
by putting a in the first m and 1- a in the remaining n, for various a. From this
point of view we can see that if with our convention on binomial coefficients
the above equalities will still hold even if we change our limits on a and j to
ensure that all the terms make sense for arbitrary values of and i.
a
This calculation is very similar to that following proposition 2.1. It is reasonable to expect that in calculating Pm,n by this technique we would be able to
use the trick of taking real parts of powers of a primitive mth root of unity in
the same way to complete the calculation.
The expression on the left hand side in this lemma is close to what we have
when we multiply the part of A corresponding to each by a vector like vo, of
appropriate dimension. The difference is that the sum in the lemma includes
all the terms corresponding to elejek with j < 1 as well. These columns of
A are present but are in a block corresponding to the smaller value j. The
next lemma is a key observation that allows us to rearrange the triple (1,j, k)
without affecting the formula above.
Lemma A.4 Let 1+ j + k = 3n. If p is the differencebetween 2 of 1,j, k then
R(wP) is independent of which 2 we choose.
Proof
We simply note that working modulo 3 we have just 0 and ±1 to worry about
59
and R(w) = R(w-1 ) so the order of the pair that we choose does not matter.
Then, for example,we have k = 3n- j - 1 so j - k = 2j + - 3n = I - j
0
modulo 3.
Let us now consider how the product Ac + Lv relates to what we obtain by
summing the formula in lemma A.3 over 1;. If we sum over I from 0 to 3n
and for each for j from 0 to 3`1J we obtain an alternating sum of binomial
coefficients which of course is zero. In this sum the sum of the coefficients of
terms corresponding to etejek is -3 if I -j is not a multiple of 3. This follows
from the previous lemma. If - j is a multiple of 3 then the total coefficient
is 6, 3 or 1 according to whether 1,j, k are all distinct, 2 are equal or all are
equal respectively. Note that the coefficient 3 when two are equal consists
of a 2 from 1,1,3n- 21 and 1 from 3n- 21,1,1, which is how we modify our
coefficients to make the formula work. Now what about Ac+ Lv? Our formula
for the coefficient of elejek is that it is 1 if two of 1,j, k are equal, and 2R(wl - j )
otherwise, except when j = n, in which case we have to add (1)n as well.
There is also the exception that the coefficient of enenenis zero. Putting this
altogether we see that with the coefficients as we claim we have the formula
3(Ac+Lv)i= 3 (-1)n-- E
+2 (zi
(l
(3n-2i (3 2i + a - )
J
(2(n
3n-2i)a)al-Ja - +J+a))i a
=0=O
In fact by lemma A.4 if we sum of all
and rearrange the terms we arrive at
3 times the sum from = 0 to n minus twice the term with = j = k = n.
This sum is zero on the one hand and 3(Ac + Lv)i minus 3 times the j = n
terms minus twice the n, n, n term on the other. Taking 3(Ac + Lv)i to the
other side gives the formula above.
If we denote the expression on the right hand side as 3x + 2y then we see that
y is 6 times the expression for the ith component of d (unless i = n) so it
remains to show the identity 2x = -y. We remark that if n > i each term in y
is even so this is certainly reasonable. For n = i we instead want 2x = -y- 1.
At the moment the binomial identity that we have to prove is somewhat difficult but it will become clear once we see an alternative expression for the term
labelled x. This is supposed to be the sum over of the coefficients of x i 3n - i
in elene2.n-iland was arrived at by first considering the terms from el. If we
first consider en and so reverse n and in the formula we instead arrive at the
expression
i 3n-2i 2(n-i+a)h
=0L-1
1=0
-i+a
a=O
60
J
L a=O
a)(n-a)
a
n a
Weobserve
that
2 -il=i-aa(1)- '1)
Weobserve
t at.2
EZ( 1)n
Il=i-a
+aa)
(n - ii+a
(n-l-(2(n-i+a))
= (1 _ 1)2(n-i+a)_ (2(n-i+a)) O
1-i+a )-'(-I-\n-i+a
!s
if n > i we have 2x = -y as required. If n = i then the same holds for all
a except a = 0 since 0 is not well defined. However if we look at the a = 0
term we have 2x = -2 = -1 - 1 = -y - 1 as required. Note that here only
the = n terms contribute. If i > n we should have a similar problem when
a = i- n. Howeverin this case we still only get a differencefrom the = n
terms and having a = i- n means that all i of the numbers 1 through i must
go in the third set. However this set has only n members since = j = n so
in this case the problem goes away and we have 2x =-y as required.
This completes the combinatorial proof of the formula for P3 ,n.
61
B
The canonical differential
In this appendix we calculate the modular forms a2i- 1 (in the notation of
Katz) that occur as the coefficients of the differential on an elliptic curve, as
polynomials in g92and g3. Katz gave formulae up to all, which we will extend
to a1 ol, and indeed we will note a correction to the formula for all.
We recall that in Weierstrass form the curve is given by an equation
y2 4x 3-g 2x - 93
and the canonical differential is w = dx/y. If we define t = x/y then the
modular forms a are defined by
w = -a
i -l
dt.
To calculate this series we introduce s = 1/y and divide the Weierstrass equa-
tion by y3 to obtain
s = 4t 3 - g 2 s2 t
g3s3.
This formula can be recursively substituted into itself to determine s as a
formal power series in t to whatever order of t we require. Indeed if we define
s = and = 4t3 -g2slt
- 9g3s
3_1 for n > 0 then we claim that sn-s n-1
l
mod tn . This is true for n = 1 since s = 4t3 . Now if sSn-1 mod tk
then s m
l mod tk+3m because the smallest term occurring is t 3 . This
means that the definitions of sn+l and sn agree mod t k+4 so they are equal for
4 extra terms. The conclusion is that the sequence sn converges to a unique
power series that is s and that sn s mod t 4 n + 3 .
To determine the differential in terms of t we first calculate that
dx/y = dt + tdy/y
from the quotient rule. Also ds = -dy so dy/y = -yds = -ds/s and we
obtain
w =dx/y =dt - tds/s.
Next we differentiate s = 4t 3 - g2S2 t - g3S3 to obtain
ds = (12t 2 - g2s2 )dt - (2g2st + 3g3s2)ds
and we substitute this in the above, which gives
w = dt-
t(12t2 - 92s)
dt.
s(1 - 2g2st+ 3g3s 2)
We can replace s in this formula by the power series calculated above and then
formally invert the denominator to obtain the series that gives the formulae
62
for the ati. We note that we have to divide out by t3 before performing the
inversion. This means that if we use sn instead of s we will obtain a formula
for w that is accurate mod t4n .
We carried out this calculation with the aid of Maple and found that the first
50 terms of the series are:
-2 + 169 2 t4 + 96g 3 t 6 - 192g22t8 - 2560g39g 2 t0l° + 2560(g2 - 3g2)tl2
+53760g 3 9g2t4 716892 (592 - 48g32)t16 - 34406493(392 - 2 )t 8
- 20g2)t 2 o - 27033609392(-792g3 + 16g2)t2 2
+516096g2(g3
+1081344(24092g3g - 7g6 - 60g4)t2 4
-337379328 g2 g3(g3 - 52)t 2 6 + 18743296g2(6g - 315ga23 + 28Og4)t2 8
+ 45g6)t 3 °
+13120307293(489g4 - 400g2g392
-56229888092(3g 6 + 448g4 - 22423g2)t32
2 + 55g26)t
34
4
-1853882368g 2 g9
3 (336g - 77092g3g
+463470592(201609g4g2 - 1344g 6 + 55g9 - 5544gg26)t 3 6
- 2242g3g2 + 1lg26)t3 8
+158506942464922g93(224g4
4°
-270952038492(109200gg492 - 26880g6 - 18720g2926+ 143929)t
-325142446080g 3 (4704g493g2- 2548g2g + 91g 9 - 192g6)t 4 2
+455199424512922(-10560g6 - 2145g2g 6 + 13929+ 18480g4g3)t 44
100809392 + 3360gg26 - 91929)t4 6
-5484069257216939 2 (1536g66 - 2217609g4926
+ 18480g2g 9 - 9g1 2 )t 48
+9971035--)01312(6336g8 + 23654492g3g
9
6
-23930484031488092g93(35g - 2640g + 73929g4923- 1650g 296)t 5 o
Note that the coefficient of t2k is a2k+1. We calculated as far at t 124 which
requires determining the 3 1 st iteration of s. For the purposes of section 4 we
include these terms below.
6
2 + 82368g 8 - 81681692g3g
+ 466752g4926- 29172g2g92)t5 2
1173062942720092(119921
+18299781906432093(-190080939 6 + 3520g8 + 2827449g4926
2> 9
1)5
2 54
-44880gg
2 + 765g~)t
-253381595627520g922(320320g8 - 24752gg29 + 85g 12 - 1345344g2g 6
+510510g
49 )t 5 6
6 + 117374494g 6
- 12031921383014409392 (91520g 8 - 130416092g3g
-138567g299
+ 1938g1 2 )t5 8
-257826886778880(256256g'
- 11411400g929
4
- 1292g
5
°
+ 440895g; 2 g2 + 42602560926g6
- 19219200g3g8)t
60
+ 256256g 8
+399631674507264009g2293(91291294926- 153753692g3g6
2 6
-8299293g g9
2 + 969g )t
-532842232676352092(969g 5 - 61501440g6g 6 - 383040g2912
+12172160g49g + 461260809g892g3- 23429129g°0 )t 6 4
-34101902891286528093(20134409g8923 - 63423369g6g6- 19968g T0
63
+2642640g4g2 - 190190g2g 12 + 1881g15)t6 6
+1406703494265569280g2(57g 5 + 7468032gag23- 6534528g6g6
-2584Og3g~2- 905216g ° + 990080gg)t
°
+630143857773772809293(-22364160g
68
+ 589388800gs92g - 111141888Og3g9
168245g 5 )t 7
+340372032g4g29-259471000259788800(-358465536923g ° + 4807g 8 + 112720608g 2 g4
6 + 1545882624g26g38)t
72
+2715648g12 - 2477376g25 g 2 - 944706048gg29
19835200g]g 2 -+
+19749449848345067520g3g
-25049024gg +
(7920640g]
1204280gg92
-
°
- 8712704093g9 + 10735296093gg92
8855g25 )t74
5
+43887666329655705692(-14277943680gg2 + 31925836800g6g8- 25741485gg92'
+13728792Og49 2 - 124156032OOg°0 g3 + 361181184g32 + 44275g1 8 )t7 6
5
2 - 17892160gg92
+2501596980790375219293(115115g8 + 292454493g
+444050880g3g 2 2 - 494247936gl 0 g2 - 2427110400g36g29
8)t 78
+2831628800g
-34232379737131450368g2(555663360gl2
+ 8855g 8 - 7945986048g° 0g3
+1330243200gg268 - 4514764800g g + 356592000gg9392
-5768400g2g2 5 )t 8 °
-9596769704084714291209293(-172931616Og3g9
+ 2766905856g6g8
5
8
+253955520g34g22 + 49335g - 8691930g2gl + 18522112g 2
-812657664g ° 2 )t8 2
+95967697040847142912(16855121920g3g2 - 79380480g34
2
-385165872Ogi
g6
125149280256g6glO ° - 35790300g
+ 25395552Og 5 g4 + 49335g2l)t 8 4
-
8
g2
+14526255744O29g
+173317660855769940099072gg2293(-240787456g3 °Og + 30232800g4g9 2
8
+ 532097280gg s - 251536896gg9+ 4485g~
-889200gg
5
+ 13230080g1
2 86
)t
+569187720380196847616g2(-8438730300g3g 5 + 104504400gg2918
-712903935744gg899 + 151222046976glg 2 2 + 844923183104glog6
+3492741120g3 4 - 192027996160g3g2 - 130065g2l)t8 8
-5008851939345732259020893(-56408625g32g8 + 2209413024g34g5
-39278453760g °9g + 60008748800g8g9 - 22094130240g93g2
3~~~~~
2 1
- 15876096g 4 +4167475200g32 g3 + 254475g2)t9 °
8 + 113109g1- 2677768192g 4
+102453789668435432570880Og2(-10015005gg
2
+60919226368g1
2
+913368456gg915
g3 - 173511010816ga°g6
+
-
19110478464g6g12
110416097792ga8g)t92
+79392506545934553841664g2g93(2629575g2 1 + 116931333376093g929g
0
-343434685440g36g1 2 + 28977301584g4g 5 - 1055681003520g °g96
-647927280gg9 8 + 189481205760g 2g3 - 2794192896g3 4 )t9 4
-20711088664156840132608(15107968139264g26g32 + 14669567278080g 2g8
+87121298880g2 8 g4 - 2106399404032g 5 g36- 2967636598784Ogga 0° + 876525g24
-1285328732160g39 4 - 851561568g2lg2 + 401665228893g6 )t9 6
590155005440 g2
-423860771132189704192 3(736281 21 ++ 59015500544093
-4639283860771132189704192g22ga(736281g
3
21234
64
-14105353152Og36g
2
- 708186006528g3°g
6
+ 10164151536g934g92
+196815962112g223 - 70291415049g14- 200608254gg328 )t98
1 - 35263382889Og3g5
+270624891878316044399411292(-1 116558OOgg32922
+1273703200094928 + 291834892800g83g
2
4 g23+
+9128755200g6 - 753122304000g
1051839224
-741909149696093g°929)tl
+ 5221647974400g2g
6
°
6 - 866118176g2g22l
19330349419879717457100800g93(451067904gl
-173811499008g34923+ 4013054036992gg22 - 6213761089536g°g 9
-793790908416g6g 5 + 2888487g24+ 49368736032g4g 8
+2509403516928g]
2 926)tm
2
-78867825633109247224971264g22(-1661433446400g 4 g + 48865689600g]6
+8279070800gg28 - 260362502400g'g2 5 + 56637g24+ 252923573760093g12
- 65447200gg 12l)tO 4
-7919499427840g3°g9 + 7424530713600g29g26
-8523910880190160209883168768g392(-35150900gg221 + 322191360g16
2g926
+2239960800gg98 - 485450085120gOg99+ 27158046720093g
+106981g24
- 31950643200g4g23
2 )t0 °6
-41343276480g6g95 + 2489487616009gg89
+(-7353299750441951439453984452777292595200g34g2
-4353255581424835834815640101383779123209g36g8
+4910953761902303282778196759533406126080g8g5
+22570545067328767612768480056441411993600g]2g
-18620699680546233280533996046564164894720g3°g 2
-915443479669088258880047862157148160g38
+4201885571681115108259419687301310054409g36g2
-87932877737813697479160016089907200gg3224
g2
+12274216864919649910194473970067046400g
8
+70145885374894117647193175228416g227)tlo
+88517536063513202179555983360g
3 g(-
152990329856093g°92g
+ 1142235494400g3 2 g 6 + 168113g24 + 4270858592g4g
8
1
-89747074560g6g95 + 5119262720g 16 - 60288800gg32
+635708444800gg89 12 - 208865918976g149g)tllO
-194933512852581626421444608092(189688230051840g34g26
-4671141995520g69g8 + 117536852160g34g
2 - 766544688gg24
-274323429918720g °g92 + 419553481052160g]2 g + 18441911255040g36 g3
-156370206720g38 + 565471g27+ 60620579379360g3g2 5 )tll2
7 - 18354092630016g4 g6
-59543327562243114979641262080g3(407529192g
+68827847362560g1 29g
- 711742512499203g 912
+875393925120g16g9 - 2957446566360936g8 + 124524065952gg
+24376529274240g 8g2 5 - 1589363490gg24 - 16166092809g18 )tl 1 4
1567485612g2g24
+16161760338323131208759771136g2(- 11641361691960g6gl 8 - 3281716838400g 8
+1738150135787520g32g
+262933070400g34g
2 - 919214014118400glog22+ 172352627647200ggl
65
5
+158999180820480g1692g + 1072445g27 - 1050530301849600g 4g46)t" 6
+970272699609504473620630470656939
2
(-131398617692160g
8
-1722664944gg2924 - 34819276800ga + 160598310512640g3
+ 37640229026400gg39215 + 14857985142092gg
+4075291g
2
3°
2 g9
4 g6
-3962129371200g6g8 - 59526786908160g3
+4837703270400g36 g)tll8
+(-272699558638918420400406398026907648g2°
+721965202905751390557818724689134498086912000g34929g
+3844345961196332445610095594047817331507200g3g2l
-159592308010745044228570454931282362735001600g
3 6g 2
-64449329349467926294051602606095761145856000938gg 8
+400690687612692021759589392202469646438236160g3g25
+6418996802845451753788655804174252900352000Og38g23
-783494421439469927773796656395614945280009gg2
1212
-920779105372600353033399865919816611764633600g
3 2
-10135258109755976453350509164485662474240g32
+428578145060906627158316119737772277760gg3222 7 )t 2 °
2
~ ~ ~~~10 g 12
- 122455106226578840463845086986240gg2(-30242538992640g
3
3
2
+7338263137920g8g' 5 + 45772491448320g]2 g9 - 675892657440gg18
+22707856080gag 1 - 22714168688640gg4 6 + 2877813227520g6
240179247gg224 + 525844g227 )t 2 2
-50487951360ga8+99556996932177919076296818688g2(14726735444678400g93g 8
°
+290884183154073600g3 2gl 2 - 105705234414024960g]°g° 5 + 4311920892g3
6g 6
-6096167686963200g38g92+ 8882987201003520093g
°
-288697084032614400g4 g + 37563035811840g93
1
4
+14430842538840gg492- 780963243278400g6g
-72682814985gg327 )t 24
66
C
Tables of P, and Pm,n
Table of Pn
n
Pn
1
elfl
2
ef2 + e 2 fl2 - 2e 2 f 2
e3 e+ ee 2flf 2 - 3 1e2 f3 + e3 fl3- 3e3flf 2 + 3e3 f 3
3
4
5
-2ele 3 fl + 2e4 f22+ 4e4 flf 3 - 4e1 2f 4 - 2e2flf3 - 4e4 fl2f2 + 4ee 3f4
- 4e4 f4
2f 4 ±
+e~e2flf 3 + ele 3ff 2 - ele3 flf 3 + e 4 + 2f
-5ef 2 f 3 + 5e5 fx2f3 + 5e5 flf 2 - 5e 5 f3f 2 - 5e2 e3 f 5 + 5e2e3f 5 + 5e2f 5
-5ee 2f 5 - 5 5f 1 f 4 - 5ee 4 f5 + 5e5 f5 + el5 + e5f 5 ±ee
+ 2 flf 4
+
2e
f2f3
2ee
f
f
ee
flf
ele2f
f
+
-3ele2flf
3
3
2
3
4
2
3
4
3el - I 3f~3+eC
61e3fl
+e 2 e3 flf22 + 5e2 e3 flf
4
- e2 e3f 2 f 3 - 2e2 e3 f2f3 + e1e4 flf
+5ele 4 f 2f 3 - ele 4f2f3 - 3ee 4flf
6 eL4e
2 fl f 5 + e2e2f 2 f 4
-
4e2ef
f2 +
1f5
2
4
+ ele4f3f 2
+ e3e3 ff2 4
23e 3 f2 f 4
+
ef 6 3f - 2e/f 6 ± f + 2 f32
3ef2f 4
2e2f + 2
f l
+3e'f 6 + e6f16- 2e6 f23+ 3e6f32- 6e6f 6 2e2f2f4
-3ef 2 f 4 - 3flf 5 + 3e2e4 f32- 2e2 e4 f23- 3e2e4f32+ 21le5 fl
-e 3e3flf5
-
3eie2e3
-3e l e 5 f32+ ele 2 e3 flf 2 f 3 - 3ee 2 e3 fl2f4 + 4eIe2 e3 f 2 f 4 + 7ele 2 e3 flf
2
3 _-e2e4f f 4 + 2ele4f2f4 + e2e4 ff 5
+e 2 eIf2f22 - 2e2 e4 fl3f 3 + 2e 2 e4 ff 4 + 2e2 4 f2 f4 - 6e 2 e4 flf 5
-3e2flf 2 f3 + e
ff
5
+ele 5 f 4f 2 - 4ele 5 f2f2 - ele5 f13f3 + ele 5 fif 4 - 6ele5 f 2 f 4
-ele, 5 flf 5 - 12e1 2e3f 6 - 12e6 flf 2 f 3 - 3e2e4 flf 2 f 3 + 4e2 e4 flf 2 f3
+7ele 5 flf 2 f 3 - 64e 2 f6 + 9e2e2f6 + 6e1 3 f6 - 62e 4 f 6 + 6e 2 e4 f 6
+6ele5f 6 - 6 6 ff 2 + 9e6 f+2f2 6e6 f3f 3 - 6e6 f2f 4 + 6e6 f 2 f 4 + 6e 6 flf 5
3
+ 4l7e f27 - 777 flf
7 7e4e3 7
7 f f 4 - 7e 6 f7
+7e2e 3 f 7 - 7ee3f 7 - 73e 4 f 7 - 7e2 e 5 f 7 + 7le21f7 + 7e 7 f4f 3
366
1
7616f
2
f
-766f
-7e 3 e4 f 7 + 7e2e5f 7 - 7e5e2f7 + 14e7 fl3f + 7e7f22f3 + 7e7 flf32 - 7e7 f 3 f 4
63 2f
e3 2 f6f6
-f e361f2/f5
-5e3e2flf
+e5e
2f5
6 eeef3f43f4 -3eef
2f5-+
2flf6 + eef
3 ff 5 - 2e4 3 f 2 f 5 -4le 3 flf 6 + e2e3 flf32
6 + e4le
-e2e 3 f 3 f 4 + 2e e3 f2f 5 + 3e2e3 f2 f5 - 7ee2 3 ff 6 + e1 e23f2f
3
3
2
22
2
-2ee2f l f32+ 5e1e2f3 f 4 + 4ee2ff 5 - 7ee2f 2 f 5 - 4ele2flf 6
+5eeflf
+elf 7 + e7f 7 + 7e7f 7 + ee 2e 3 flf2f4 -3e
2e3f 3f 4 -4ele
2e 3f
+6e2e2 e3 f 2 f 5 + 9ee 2 e 3 f1 f 6 - 2e2e3 f lf 2 f 4 - el e2flf 2 f 4
- ee 4f2f 5 + 2e3e
4 f 2f 5 + ele4 fl f6
+ 5e 3 e4 flf32 + 3e 3 e4 f3f 4 - 5e3 e4lf 3 f4
+ele.l
4f 3f 4
4 f3f 4 + 3e3e
+e3eflf2
-
3 e4 flf
3
-7e 3 e4 f f5 + 7e3e4f 2f 5 + 7e3e4flf 6 + ele 5 f f3 + 2e1e5f.f 3
+4eje 5 flf3 -ele 5ff 4 - 7ee 5f3f4 + ee 5ff 5 - 2ee 5f 2f5
-e2e5flf 6 + e2 e5 f3f2 - 3e2e5 flf23 - 2e2e 5 f4f 3 + 3e2 e5 f22f3
-7e 2 e5 flfl + 2e 2 e5 f3f 4 + 7e2e5 f 3f 4 - 2e 2e 5 f2f 5 - 3 2 e5 f 2 f 5
+7e265flf6 + ee 6 f 5f2 - 5ele6 f3f22 + 5ele 6 flf23 - ele6 ff 3
67
2
5
+ e1e6f3f 4 + 7ele6 f3f 4 - ele6 f2f5
+7ele 6 f2 f5 + ele 6 flf 6 - 21ee 2 e3f 7 + 14ele2 e4 f7 - 21e7ff2f3
+14e7 fl f2 f4 - 3e3e4 flf2 f4 + ele2e4 flf 2 f 3 - 2ele 2e 4 f22f3
-ele 2e 4 flf32 - 3ele 2e4 f3f4 + 8elC2 e4 flf 2 f 4 - 2ele 2 e4f 3f 4
+3ele 2 e4 ff5 - 4elC2 e4 f 2 f 5 - 8e 1e 2 e4 flf 6 - 3e3 e4f f2 f3
+
-2e3 e4flf 2 f 4 - 4e2e5f2f 2f 3 + 3e2e5 flf2 f4 6e2e5 f2f 2f3
-4e 2 e5 flf 2 f 4 + 9ele6 f2f 2f 3 -8ele 6 flf 2 f 4 + 7e7 f2f 5 - 7e7f 2f 5 - 7e7flf 6
-7ele 6 f2f-ee2f
3 - 4e 6ff
8
2
2 f 8 - 8e2e6 f8 + 12e8 f2f32 + 8e8flf 7
-8e 2 e2f8 + 8e8 f 2 f 6 - 8e61e
2
+2044ef 8 + 2Oe8 fl f - 16e8 f f - 8ee4f8 + 8ele 7 f 8 - 8e8 flf
2
+8e8f1 3f
5
+ 8e2e 6f 8 - 8esf2f 6 - 8e8f22f4 + 8e8 fl5f3 - 8e8f 2f
2-4ele2
+e-le2
le~flh
+ 8e3e f
5 8
4 f8
8 - 8e4e
8 - 16e2e32f
-8e 8 fl4f 4 + 8e 8 f 3 f 51 + 12e2e2f
21
2
~~~~~~
1~~~
1 3
2e2e2f42
+
f
2e42f
f
+ 2e4f 2 6
l 7
+8e 3 e5 f 8 + 8e5e3 f 8 - 2e4f 3 f 5 24
3
2ff
21
2
2
1~2~ + 2e4f2f32
- 4efj3f 5
4ef22f4
4ef2f32
+
+4e2ef42 -3 4e2e4f4
4 _
2
34
-4e4f 3 f5 +4eff 6 - 4ef 2 f 6 - 4e4flf7 - 4e e5 f42- 2e3e5f24
-4e 3 e5 f42 + 4e2e6 f42 + 2e 2 e6 f24 - 4e 2 e6 f42 - 2e l e7 f24 - 4ele 7 f42
3 f 5 -4e e3f 2 f6
7 + e2e3f
+e64e2 flf7 ± e'e2f 2 f 6 - 6ee2flf
1
6ele2
e 3 f2 f6 - e5e3 flf 7 + e8lf8 + e4f42
+9e2e~flf 7 + e5e3 f f 6 -2e
8 f8+ 2e f4 +ee e flf 2 f 5
-J6,
3
+
1
+9lflf
2 3
8
+2e42f
8 + e4f + 6e4f4 + 4e4f8 + e8 fl +
e
-3e3e 2 e3 f 3 f5 - 5e3e 2e 3 f2f6 + 8e 2 e3 f 2 f + 1e3e 2 3 f 1 f 7
-4ele e3f 2 + e23f22f4 ± 3e2e2f3f5 + 5e2le3f2f6 - 9e2le3f2f6
2e2ef 22f4 - 7e2ef 3f 5 - 5e2ef2f 6
-5e2e2flf 7 + e2ef 2f
+2e2 e2f 2f 6 + 82 e flf 7 + e4e4fl3f5 + 3e4e4 f3 5 -e 4 f2f 6
-
e~~~~2
f2f3
-2e-e 422
+2ele
4f f 2f
2e4f4 +ee e2
7 + 4ele
4 f2f6 + ee4 fdlf
- 2e2e f f -4e2e f f
+4e2e 4 f22f4+ 2e2e4 f1f
5
+ 8e2e 4 f 3 f 5
4
4 2 6
6
2 f4 -8e2flf
- 8ele 3 e4f42- 4e~flf22f
ff 72e~~flf7
3 + 4e4f1f
+8e2e 4+8
43
4
+8e]fIf2 f5 + e3e5fl4f4 + 2e3le5 f2f 4 -ele 5fl f 5 -3ee
5flf 7 + e1e2e3ff 3f 4
+e3e5ff6 - 2e3e5 f2f6 - e3le
3f 4
5 f3 f5
-3ele2e3 flf 2f 5 + 6ele2e3 f3f5 + 5elee 3 f2f6 - 17ee2e3flf 7
-2e2Cflf3 f4- e2e23flf
2f 5 e2 ef l f3f 4 ± 5e 2 fflf 2 f5
e4f2f 4 -ee 2 e24fA
4 flf 3 f4
e4 f2f2f4 - 2e2e
-3ee 4fif 2f5 + e2e
2if6f
gf22
f
I je22
flf5f+
23f
-4ele 2 e4 flf5 + 11ee 2 e4 flf 2 f 5 - 9ee 2e 4 f3 f5 + 4e2le2 e 4f2f6
-6e2 2e 4 f 2 f 6 - 1Oele2 e 4 flf 7 - 2e2e4 f f 2 f 4 - 4e2e4 flf 2 f 5
+ele 3 e 4 flf22f3 - 2ele 3 e4 f2 f - ele 3 e4 f 2 f32- ele 3 e4 f2 f 2 f4
+10ele3 e 4 f l f 3 f4 + 4ele 3 e4 flf 5 - 10ele 3 e4 flf 2 f 5 + ele 3 e4 f3 f5
-9ele 3 e4 f2f6 16ele 3 e4 f2f6 + 9ele 3 e4 flf 7 - 4ee 5f2f 2 f4
2
+4ele5 flf 3f4 + 3ele5 flf 2 f5 + ele2e5flf2f3 - 3ele2e5flf 2f3
-ele 2 e5f2f32 + 5ele2 e5 f 2 f32- 3ele2e5 f4f 4 + llee 2e 5 f 2f 2 f 4
-4ele 2 e5 f2f4 - 10ele2e5flf 3f 4 + 8ele2 e5f42+ e3e5f2f2
+3
2f323 ee5fl
+
7e 3 e5 f 2f3
+7e3e5 f 3 f 5 ± 8e3 eff6
+5ele6f, f -5e e6f2f3
-e
+8se6f3f55
e 6 f 2f6
-e2e54~f
1
1
3e 3 e 5 f 4f 4 + 8e 3 e5 f22f4 - 3e 3 e 5 f3f
8 3 e5f 2f 6 - 8e3 e5 flf 7 + e2e6 f5f 3
-ele6fl'f4-2ee6f
2f±+
+ 2C2 e6f2f6
1
68
f4 + e2e6flf5
- f4f22
f
eeff++ e2e6
e426flf7
1~~~6
1
5
-4e 2 e 6 fIf3 - 2e2 e6 fl5f 3 - 9e2 e 6f2f32f + 2e2 e6 f 2 f32+ 2e 2 e6 fl4f4
-4e 2 e6 f f4 -2e 2e6fl3f 5 - 8e2e6f 3f 5 + 2e2e6 f2f6 + 4e2e6f 2f 6
-8e,2 e 6 flf 7 + ele 7f1 6f 2 - 6ee 7 f4f 22 + 9e l e 7f2f23 - ele 7 fl5 f3
-5eie 7f2f + 8ee 7 f 2 f32+ e l e 7f 4 f4 + 8e 1 e7 f2f 4 - ele 7f3f 5
-8e.e 7 f 3f 5 + ele 7f2f 6 - 8ele7 f 2f 6 - el e7flf 7 - 32e3e2e3f8
+24elee3 f8 + 24ee 2e4f 8 - 16ee3 e4f8 - 16ele2e5f 8- 32e8fl3f2 f3
+24e 8 f l f22f3 + 24e8 f2f 2 f 4 - 16e8 flf 3 f 4 - 16e8f 1f 2 f5 + 3ele 2 e 5 f3f 5
-8e:e 2 e5 f l f2 f 5 + ele 2 e5 f 3 f 5 - 3ele2 e 5 f2f 6 + 4ele 2 e5 f 2 f 6
+9ele 2 e 5 flf 7 - 3e3 e5 fl f2 f3 + 6e3 e5 flf22f3 - 9e3e5ff 2 f 4
+e3 e5 flf3 f 4 + e3 e5 flf 2 f 5 - 5e2e6 fo3f2f 3 + 5e2eofl f2f 3
+4eie6ff 2f 4 - 9ele6flf3f4 - 3e+eoff2f5 +8e2e6flf2f 3
2
-6e 2e6 fj f 2 f 4 + 16e2 e6 f l f 3f 4 + 4e2e6 flf 2 f 5 + 11el e7f13f 2 f3
-17ele7 flf22f3 - 1ele 7f2 f2 f4 + 9el e7 flf 3 f 4 + 9ele7flf 2 f 5
+4efi- 8e8f8
-3e4e 5 f33+ 9egf22f5 6f 3 f 6 + 6e3 e6 f - 9eflf 8
-30c ef 9 9e2 e7 f 9 - 3Oegff
-9e3flf
+ 3ff
9 e 7f
+ 3eff
7
-9ele 4f9 + 9egflf24+ 18egfl3f32- 9e9f 2f 7 - 3ef 2f7 + 9e4ef9
+18eef 9 9e3e6f 9 +
+ 93f
9e
93ee4f3+
- 9e9f2f 3
+9ee 5f 9 - 3e3f 4 f 5 - 9e9flf 4 - 9e 4e5 f + 9e9 f 4f5 + 27e5e2fg
+3e,'lff2- 3eflf8 + 9e9f2f 7 + 9egf6f 3 - 9eae 6 f 9 - 9ere4f9
-11e
l fie 3e4f 7 - 9ele 8f 9 - 3e2 e7 f - 9ef3f 6 - 9ef 4 f 5 + 27ef 5 ff22
+3ee 7 f3
9e7e2f4e2f2e
e f 2 e5 f 7 -11e2f2e
7 f 3 f 4 + 9e 9 flf42
2l247+
+l~f
%f
+
f72
e fe f2f
2fleff ± 9e1f e2 f -- 3ee
3ff f f
-9e,'e 3 f9 - 3ee 8 f33 99e
7
3
3 7
5 3 6
+3eIe5f33 - 2e4le5f2f7+ e6f2e3f7 -e2f5e7f4
- e3f2e6f7
-e5 j e4f7 + 6e3f2e2f7 + e2f,6e7 f 3 + e4f]2e5f7+ 6e2f 3e7f32
+e2je7f 7 +efe
4f 6fl/ ef2
3fle4f 4 ee 3f2f3e4f4
6
e4
-efal
6
-e2 f12e1 f 3 e 4 f 4 - e2 e3 2f 3 e4 f 4 - 2e 2f2 f 3 e4 f 4 - 2e 2e 3 flf2e 4 f 4
+e 2 e3 f2 f32e4 fl + e 2 e2f 2 f 3elf 4 + 8e2 e 3 f2 f3 e 4 f4 - 2e2fl2e3 e4 f 3 f 4
22~
-eIJe
3e4f 2f5
±e3Lfe6f4- e
ef2 e¢2fef
+ 5efl3e 3e4 f 6 + elf1 e5f 5 - e4f3e5f 6
1f66 f 5 + elfe6f6
+ elf'e7f5
-
elfle7f6
-4e f2e5f2 f 5 -e2 fe 2e5f 3 f4 -4e2 fe 2e5f 5 + 15e2fe 2e5f 2f5
+4efle
2e 5 f6
+ 5efle
6f 3f 4
+ 4e fe 6 f 2f 5 - 6e fle 2e3f 7
+5e f1 2 e 4 f 7 - 6 fe 7 f 2 f3 + 5ef1 3e7 f 2 f 4 +
fle 2 e4 f 2 f 5
-5elfjeC 2 e4 f 6 ef
_ e2 e 5 f2 f 4 - 5ef1e 6 f 2 f4 - 4e2
f2e 7 f 2 f 5
-eClfe2e4 f4 -eifle~f2f
ef1 e2 f 8 + efle 6 f8
4 - 5elfle4f
-6elf3esf 2 - elf2e8 f 7 - 7e5fle2f8- 7elfl5 8 f22 + 14elfe 8f23
-elJ'fe
f
8f2 - elf4e8f5 +elfle8f 6 - efe 8f3
57 8 +elf7e
+eClj
e8f4-6efle2f
efef
e]fe
4f4-e4fle
+el~ef4
5f8
- 66e l 8 f 14e
l2fs e8 feaf
1fleJf8
e415ef8
9
.
-ef'le 3 f8 - 5efl 3 ef 6 + 5elf2e~f 7 - 5e3f le6 f42+ 5e2fle 7f42
+5e2fe 3 e 4 f42+ 5elf2e~f 3 f4 + 19elflee2 f 8 - lOefle f2 f 6
+19elfle2e 4 f 8 - 10elfle 2 e6 f 8 + 19elfe 8 f22f4 + 19elfle 8 f2 f32
-10~fle 8 f3 f5 --lOelfle
3 e5 fs-3eifle
2e 2f42
2 f32
-10elflesf3f5
10Clfle3¢5f8
- 3elfle2
-- - 3elfe2f
3ef 4~f
69
-6eifie
f3 f5 + 14e l fl e2f 2 f 6 -6elfle
-7elfle42f8 - 5elflef
3 ef42
+ 14elfle 2e 6 f42
5elfle 8f2 + elfle8 f8
e3ef2 f 6 ± 4e fle 2e f 3 f5
4
8 - 7elfle 8f2 -
e4fle 2e3f2f6- 2e3f l e3f 3 f 5 -4f 3f 5
2elle
4 f 2 f6 + 2elfle2e
2 e2f7
7elfle2 e2f2f6 - 12elfe
1
3 7- - 3elfle
+7elfle2e3f
2
f 2~~~~2
2
2
f3
+5elflee4f6 - 12efleie 4 f2f6 - 5el ee 4 f7 + f 2ef 2f5
+4e4fie5 f3 f 5 + 3e4fle 5f 2f6 + eIfe2e 3 f 3f 5
-
elfle2 e4f 3f5
l
3e-f2ee 4f2 f5 + 5e2fle3 e4 f 3 f 5 - 13e2f e3 e 4 f2 f6
l 3 e5f32 + 4elfle3 ef 2 f 2
-3ef 1e2 e 5f2f4 + e2fle 2 e5 f - 2efle
+14e3fle2e4f 2f6
+2elfle
-
+ 4e l f4e 3 e5 f5 + 5elfle 3 e 5 f 3 f 5 -4e lfl3e
3 e5 f2f4
3 e5 f 6
6f3f5
5f7+ 5efle6f2f4-4efe
1
5 e e f
2
3efl
f32
2 6 4
elfl e2 e 6 f3 + 7elfle 2 e6f 2
+lOelfle
3e5 f2f6 +le
-3elfle 6 f 2 f6 -12efle 2 e6 f22f4+ 3elf4e 2e6f 5 + 10elfle 2e6f 3f 5 - 3elf3e 2e6 f 6
2f - 5e2f1 e 7f22f4
+8elfle 2 e 6 f 2 f6 + 3elf2e 2 e6 f 7 - 12efef
O~~~~3elfle2ef
f,
e4
2
32f
1 2e3 f8
l e7 f2 f 6 + 13e41f e2 e 3f 8 +lOe fle 7 f 3 fs +2f,3e2lf
-12elfle 2 e4f8 + 11e21fle3 e4 f8 + 11efle2 e5 f8 + 13elfl4e8 f 2f 3
2 f4 + lle1fe 8f3f4 + llefe 8 f2f5
-3Oelf2e8f22f
3 - 12efae8f
-15el2fle 2 e5f 3f5 - 11e2fle 2 e5 f2 f6 + efe 3 e 5 ff 3 - elfe 3e 5 f2 f4
ff
+5elf2e 3e5f 3f4 - 15elf 2 e3e5 f2f 5 + e1f4e 2e6f 2f 3 + 14elf e2e6 2 4
- el f22f
5 + eef23f3
-13elfle2 e6f 3 4 - llefe 2 e6f2f5
3
3
11ee2f 4 f 5
3 f 6 + 5el e4f2 f7
-9ele2f 3 f6- 9ee2f 2 f7 + e1e4f4 f 5 - 3ele42f
-2ele 2 e4 f22f5 - 6ele 2 e5 f22f5 + 5ele 2 e5 f 2 f 3 f4 + 8ele 3 e 5 f22f5
+
-2ele 3 e 5 f 2 f 3 f4- 2ele 3 e5 f2f 3 6ele2e4 f2f 5 - 2e2 e3e4 ff 3f5
+_le3f22f 5 - 3ele 3 e 5 f3 - 3e 2 e3e4f 3 - 3e1 e2 6 f
-6e2e3 e4f22f5+ 5e2e3e4 ff2f5 + 4e2e3 e4f lf 2f6 - 4ele2e6fff3
+4ele2e6of5f + 461e2 c6 f 2f3 f4 + 2ele2e6 f23f3- 4eee 3 flf2f6
+4ei~4f~f
62ifaA
e2e
+4.
-4e2~ff
4ele 2 e4 f4f5 - 12ele 2 e4 f3 f6
2 f3 f4
° ± 3egf33 + e91f + 3e3fg + e3f33 + 7e2e2 e5 f 4 f5
+
egfl
-8e3e 2 e4 f 2 f7 + 9efg
+9e 2e2e5 f 3 f6 + 6ele 2e 5f 2 f7 + 2ele 3 e5 f 4 f 5 - 18ele 3 e 5 f 2 f 7
2eef
-3ele 2 eff5-
~~~~~~~~
22 4 f3f6-ele
~~~
9ele2e
-3eceae 4 f4 f5 + C2
2 e4f2f7 - 2e2 e3e 4 flf
-5ele2¢e3 f2f7 ± 7ele2ef 4f 5 -18ele 2 ef 3 f6 ± 13ele2ef 2f7
f ± 3elef 3f6
2 e3f2 f7 ± ee f2 f7 ± 2eef 4 5
-3e'le 2e3 f3f6 ± 1Oe~1e
-11e3lef2 f7 + ee2f 3 f6 - 5e e2f2f7 + 36ee 2e4f9 - 27e2e2e5f9
+ 18e2 e3 e4f9 + 18ele 2e6 f 9 - 27e2e 3 e4 f
+18ele3 6ef9 -27ele2e4f9
5 f4f5
+54e2e2e 3 f - 27ele 2 ef - 45ele 2 e3 f9 + 2ele 5f2f5 + 2e22e
f3
ff
+e4 e5 ff5 - 2ee 3flf 3 f5 + 2e4e 3 f lf 2 fo- 3e2e5 2
+8e2e5 flf 3 f5 - 6ee 5 f2f2 f5 - 6ee 5 f2 f3 f4 + 4ee 5 flf2f6
+6ee 5 flf2f4
-
+2e4 e5 ff 3 f5 f2f 5 -3e
3 e62-9e
-941
eaefx9s
4e4e 5 fl2ff 3 + 7e 4 e5 flf 2 f3
+ 3e e flf
4
3 6
6 ff 5
4e4 e5 f f 5 - 4ee 5f2f3 - 2e3e
ff
2ee 7 3
3 e6 ff5 ± 3e3 e6f+ f3 + 2e1e7 f3f 5 -
13e4 ef2
3
3 f4 -
e3e~eff f9
70
+ 2eie~f22fs
-
2 23
+2e2 e7 flf 4 + 5e2 e7 f22f5 - 2e 2 e 7 ff 5 - 5e2 e7 f23f3 - 2e2 e 7 f6f3
-9ele 8 f2f 5 + 7e4 e5f 2f 2f 5 + 2e4e5 f 2f 3 f4 - 18e4e5flf 2 f 6 - 3ee 5flf2f 4
f2f~f4
f22f
-5e1e 6 f 2 f3 f 4 + 9e3 e 6 f2f 3 f 4 + 9e3 e6 fl2f2f3 - 18e3e 6 flf 2 f32
+9e 3 e6 f12f2 f5 - 3e3e 6 f4f 2 f 3 + 9e3 e6 flf22f 4 + 11e e7 f2 f3 f 4
+206 2 e7 f2f 3f4 - 5e2 e 7 flf2f 3 + 13e2 e 7 f l f 2 f32- 18e2 e7 f l f3 f 5
+6e,2e7 f2 f 2 f 5 - 4e2 e 7 f 2 f 3 f 4 + 1Oe2 e7 ff4f2 f 3 - 4e2 e7 flf 2 f6
-e2e7flf22
- 18ele 8 f 2 f 3 f 4 + 9ele 8 f2f 3 - 3e3flf 3 f 5 - 3e3f2f3 f4
-3e, flf 2 f 6 - 27egf f 3 f4 + 54egfl2f2f3 - 27e9flf 2f32+ 18e9 flf 3 f 5
-27 9f2f2f5 + 18e9 f 2 f 3 f 4 - 45e9f4f2f3 + 18e9 flf 2f 6 - 27e9f 1 f22f4
-2eie
3 f 2 f7
+ e2e3 flf42 -e2e
3f 4 f 5
+ 3e2e3 f 3 f 6 - 2e2e3 fIf 7
-5e2,e3f2f7 + 9e~e
3flf 8 + 3e5e4f 3f6 + 2ee 4 f2 f7 -4ee 5 f4 f5
-e2e 5 fl f + e2e5f f32+ ee 5f 4 f5 - 2ee 5 f 3f 6 - 9e2e 5f3 f6
+2eIe5f2f 7 + 5e e 5f2f 7 - 9ee 5f l f8 + 1le4e5flf42+ e4 e5f l f24
- 11e4 e5 f4 f5 + 9e4 e5 flf 6 + 9e4 e5 f 3 f 6 - 9e4 e5 f 2 f 7
+2eIe 5 ff~
+9e 1 e5 f 2 f 7 + 9e 4 e5 f l f 8 + 9el3e6 f 4 f 5 + 3el3e6 f 3 f 6 + 2e3e 6 f 2 f 7
-9ee 6 flf42 - 3e3 e6 flf24 + 3e3 e6 f f + e3 e6 ff
+ 9e3 e6 f 4 f 5
+3e3 e6ff6 - 9e3e6f 3 f 6 - 9e3 e6f2f 7 + 9e3e6f 2f 7 + 9e3e6flf 8
- 9e2e7 f3 f 6 - 2e2e7 f2 f 7 - 9e2 e 7 flf42 + 5e2 e7 flf24
4 f 55-f
5
+e2 ,e
7fI f
-
le 2 e7f f
- 5 2 e 7lff
+ 9e2 e7 f 4 f 5 + 2e2 e7 f3f 6
+9e2 e7f 3f 6 - 2e2e7f2f 7 - 5e2e7 f2 f 7 + 9 2e7f l f8 + 9ele8f4 f 5 + 9eleC
8f 3 f 6
+9ee
10
8f 2f 7
+ 36egfl3 f 2 f 4
-
2e2e5 fl3 f 2 f 4 + 4e4 e5 f3f 2 f 4
~~~~~~~~~~~~~~~~~~~~~2
-12(,
f4 3 flO + 15e2eflo
4 - 8 22e4e 7ff6 - f21Oe2e
1Oe1 03f e26f 8 f- 2 f2e4fle
6
-8ef
3e
6 2 5-
1 e2 f2e8 f4
-
2e2fl4e 6 f 2 f 4
10eloflff
+
+ 8ele3f 4ef 6 + 2elef 4f 6
-2ece 3f42f 2 + 10e4 f2e 6 f 4 + 2e 4 f4f32e6 - 2e 4 f25e6 + 4eaflf 3 e2 f 6
-8e 4 f22ele5f 6 - 4e4 f22e6 ff 3 - 12e4 fle 6 fif 4 - 18e4 f2e6 flf 5
+28elf2 e2 eif
8 ± 2efe
6 f/2 - 4e2f23e6 f 4 + 2e 2 f 2 e8f42 + 6e 2 f 2 e8 f8
e
f
f
4ef
e
flf
2 6 3 5
2 6
7 + 4ef2e 1e 7 f - 4e2 fele 7 f6
2
~
~~~
2
-o10ele6 flo + 4ele3f22f6 - 8f22leC 3 e 4 f6 + 6e 3fe 3 e4 f 7
2 2f52
C~~~~5f8
eleafaf
4afl3~f
+5e2ef - 2e
fxe8 -2e2f
e6f 7
2 fs -15eef
3f77 + ef
2
-8e 4f f3 e6 f 4 + 12e 4 flf 3 e6 f 2 f 4 - 6e4 fj f 3 e6 f 5 - 4ee f 4 e 3 f 6
-4e¢f
e3'~6
+eJ'
7
f4- 2e'f 4 f 6 - 6e4f2f2 2 e6 f 6
1+
15eff+
62f2 2
8efee
4 f6
+2e3f2e 7 f 6 + 35e6e2flo + 4eale2 f 4 e4 f 6 - 12e2e2f4 e4 f 6
+16e3le2f 4e 5 f6 + e3f3e 2 e5f 2 f5 - 5e3f4e2 esf 6 - 5e4f3le6 f 2 f 5
-6elf 13 e2 e 4 f 7 + 4ele3 fe 6 f2 + elfl e5f 6 + ef e 6f 5 - e4fe 6 f 6
-8e3e 3 f 4 e4 f6 + 12e1 e3 f 4 e 2 e4 f 6 - 6e2e3 f 4 e5 f 6 - 10e2 e2flo + 10e4e 6 flo0
+2e2f23e2 e6 f 4 + 6c2f22e2 e6 f 6 + 12e5f 2 e2 e3 f 8 - 12e3lf2 e2e3 f 8
2f~eaesfo
~~~~~~~0ef3
-10e4f 2 e2 e 4 f8 + 24e3f 2 e3 e 4 f8 - 2e2f23e
3 e5 f 4 + Oe2f22e
3 e5 f 6
+4e4 fl4f2 e 6 f 4 + 16e4 fIf 2 e 6 f 5 + 2e4e3f4 f6 + 6e fle 7 f 3 f 4
-e 3J'3e 7 f 7
5e2e6 f52 - 5 4 e6 f52 - 5e2 e8 f52 _ efe 5 f 7 - 5e2f f33
1
62
1 1
22 4flo
2 2 e3 e 5 f8 ++ e~~f2
+25eleflo - 22elf
elf 2 e4f3 f 5e2ff
+ 2efe 2 e3 e~2
5 f4
-6elf 2 e2 e6 f42+ 9e2f2 e4f 8 - 3e2f 2 ef42 - 11ef 2 e2f8
71
-4e 4 f2e2f 6 9e4 fe 6 f - 2e 4f22e6 f6 + 5eff 6 + 8e3f 2 e2 e 5f8
3e5f 4
-8e2f e2 e5 f6 5eff 8 - 5ef 2+f8 + 2ef 2 ele5 f~ + 4e2f23e
-4e 2 f2eae 5 f6 - 5ele gf52+ 2elegf25- 8e~f2 ele 3 f8 - 28e2 f 2 ele 3 e4f8
5
+2e2f2 ele5 fs - 8e2f2e8flf 3 - 28e2f 2e8 flf 3 f4 + 2e2 fe 8flhf
~~3f
46
2f,2
2
2
3
+4e2f2ele 5 f6 - 5e 1e3f + 2e4 flf3 e6 - 2e4flf33e2 + ele2f 4f6
6 -8e2f 2 e6 flf 3 f 4
1Oelof6f 4 + 25elof 4f32
+4e2fl f3e 6 + e4fl2fe
-1Oelof22f 6
-
+lOel e9flO ± 5eef
+ 4e2f2e6flf 5
2f
-
5e 3 e 7 f52+ 2e 3 e7 f25
+ 5ee 6f52- 5ee 7 f2 - 6ee 2f 4 e6 f6
- 2ef2e 6 fs + 2e2 f4e
6 - 2e2 f7esf 3
2e2
-6e 2 f2esf 2f6 - 1
f
~~~~
e 6 -8 f6e2f2e8f42
2f1 - e2f2e6f42
e+5 6e2f3e2f7
~ ~~~
+2e2fl6e8f4 + ef2e4f42
2
2 f4e8 f32- 2e 2 f3esf 7 + e
+2el2f
2 e8 f8++ 7e f2ef
ef2 fle 8 f2
3J83f
f2 8 l-13e
+ef4ef
efe7f6
+
-6e 2 f1 e8f2 + 2ef12e4f8 + 5elf1e7f2f5 -e3fle7f5
f - 10e2 e8 f l f 9
-6e3f4e 7 f 2 f4 + O10e
2 e f3 f7 + lOe 2 ef fl 9 - 10e2 e8 f 3 7
+10e 2 e8 f3f4- lOe2 esf4f6- 7elfle 3 e 6 f33+ 11elfle 2 e7 f 9 + llelfle 9f2 f7
f 5e8sf
-4elfl 3f2 f7 + 11elfle3e6f9 + 11el fl e9 f3f 6 + 31elfl e9f2f3
5ee9 f 5 - 8e flef 9
1
4-21e lfl e e 5 f9 + 5elf l ef3 4f 5 eflegf
2
2
7
eaf9
16ef
f
e
e4f
+
f
eg
l
-elf
f
9
6
l
efle
3
e3f8
9
1
+7elf
9 7
+eIfle 8 f 9 + elf2e 9 f 8 -elf l e7 f 9 + 20efef 9 + 20elfe 9 f23
fle 5f 9
+elf8egf2 + e6lfe 4 f - 16el1fef2 - 7e1fef
3
1Oe
4 e 6 f3 f7
-7e4 flefg - 6e2flef - e7fle3 f 9 - Oe4 e6f f7
-lOe 4 e6 flf9 + 1Oe2e6 f3 f7 + 1Oe2e6 flf9 + 4ef 2e4 f3f 5 + 4e2f2e4 flf 7
+4e 2 f2 ele 3 e4 f42+ 4e2 f 2 e ff 3 f4 + 2e2 fe-flf 5 - 3e 2 f2ef2
-6e 2 fle4f 8 + 9e2 fle 8 f24- 2ef3e 4 f7 + e2f 2e2f8 - 6e2 f22e8 f 6
+6e2f2e4 f8 - 6ef 2 e6f8 + 4e2flIe8 ff4 + 28e2fle 8 f2 f32
-22e 2 f12e8f 3 f 5 - 12 2f2e 3 e5 f8 + 2e 2 f1 4ef3 f5 - 6e 2fe4f 2 f 6
+e2fe 3 e5 f42+ 2ef2e 6f7 - 9e2f4e3f6 + 4e4lf4e6f 6
+e1 fle 2 e3 f 2 f 3e4 f4 + 3e2fle4f
+ef e8f7 + elf 2e7f8 + efe8f5
-e2f16e 8 f4 + 7e4ljl2e3f8
-
1 2 e6f8 + 7ef e8 f3
- elfL e8f6 + efle8f3
-ef
60el2e2eflo - 20ele 4 e5 fIo + 1OOe3e2e 3 fIo
- 2ele 2 e7 flo
+50e4e 2 e4flo - 20e 2 e3e5flo - 40e3e3 e4 flo - 40e3e2 e 5 flo
lo + 30e2e3e5flo
-60e2e22e4flo + 30e2e 2 e 6 flo - 60e5e 2e 3 flo - 40ele2e3f
-20ele 3 e6 flo + 30ele2e 5 flo - 1Oelef 2 fs - 10ele 9 f 2 f 8 + 1Oele 9f22f6
+lOe2esflo - 2efl4e6f6 + 11el fle 9f4f 5 + 31elfle~e3 f9 + 4e5fle5f 3f6
+3e5fle5 f2 f7 + 11elfle4e5 f9 - 4efle 2e7f33+ elf4e 9f 6 - 8elf6egf22
3
+e8fle 2 f9 - 6ef2egf42 + 7e2fle 8 f 3- 8e4 f4 e2flf 5 + 3e4f 4 e2f
+2e2f 4 e2 f32 - 4e5f 4e 5 f6 - 2e 4 f 4 e2f6- 2e 4 f4 e6 f32 + 14e4 f 4 e6 f 6
te62;
+ 2e62
+3e f4e 2f 2f2
4 f3e2f 2 -e 4 f22e1 e5 fd? ++ 8e4 flf 3 e6 f23
4
-4e 4 flf 3 e 6 f6 - e2f4 e2flf 5 + 4e4f4e 6 f6 1Oe4 f 4 e3f6 - 4ef 4 e6 f32
-2e2f 4 e 6 f6 - 12elflje 2 e3 e4 f 7 + 21elfle 2 e3 e4 f 2 f 7 + 26elfe 2 e3 e4 f8
+18eIfle 2 e6 f4f5 + 1lefle 2 e6 f2 f7 + 5elfle 3 e4 f 4 f5 + 5elfle 3 e4 f3 f6
2
e~efI
3~~~~~ef
3 + elfe2e
3f 3f6 - 5efe2e 3 f2f7
-16ecfle3 e4 f2f7
-3e2fle 2 e2f 4 f 5 + 6e2lfle2 ef
3 f6
+ 9ef 1 e2 e32f2 f7
72
+el.fle 2e3 f2 f 7 + 20ele9f 2f 3f 5 - 10e3e7 f2f8 + 10e3e7f2f 8
+1Oe3 e7 f22f6 - 3fef
2 f3f -ef
1 2 e4 f3 f 6 + 17e4fie 2e 4 f 2 f 7
+elfl e2 e 5 f 4 f 5 - 19e fle 2 e5 f 3 f 6 - 14elfl e2e5 f 2 f 7 - 12e12fl
e3 e 5 f 4 f 5
+13e21fle3e5f 2f 7 - e2fe22e4f 4f 5 + 3e2fle2e4f 3f 6
-23ec2fle2e4 f 2 f 7 - 3elf2e 2 e3 e4 f42 - 3e3 fle 2 e5 f22f 5
+2e0f l e 3 e5 f22f5 + 4e2fle 3 e5 f 2 f3 f 4 + 4elf2e 2 e3 e 4 f 3 f 5
-3e.fle 2 e 3e 4 f22f5 + 7elf2e 2 e 3 e4 f 2 f 6 - 17e2f e2 e6 f22f5
+7e: f e2 e6 f 2 f 3 f 4 - 2e1f e3e5f 3 f 4 - ef le 3 e5 f 2 f5
+el2fl4e2e 6 f 2f 4 - e2f3e 2 e6 f3 f 4 + 19e2fe 2 e 6 f2 f5 - e2flef22f 5
-8e:?f ef
4f 5 - 3e2flef 3f 6 +-117e2fle4f
2e6f6
:~~~~~~~~~314¢4
?f1%f7f
4 2f7 + 4e2f4e
4
e2f
ef
+
gef2
ef2
-155,12f2eof
l9fle2 e 6 f 2 f 6 - 4elfle 2 e6 f7 9efe 7 f22f4 - 5eafl2e 7 f 3 f 5
-4e Lf1I2e7f
f 2 ee3f8 + 14efee
1 2f 6 -7e 1-1ef 1ef2 3f86 6fe
ef 1ef2e4f8
5
-13elf2e
3 e4 f 8
- 5e3fl2e2 e5 f 8 - 7e2f e 8 f 2 f 3 + 14e2f3e8 f22f3
3e2f
2
~
~~~~~
+6e]fl'esf2f4-13e9fe8f3f4e8f2f5
+19e
fle 2 e5 f 23f2
f 6 ef2e
+ 3elf
3 e5 f22f45e9f
I e 3e 5-f efee2e5f3f5
3f 5
+5elfl e 3 e5 f 6 19e2fe3 e5 f 2f 6 - 5efe 3 e5 f 7 + 5e fi2e6 f 3 f 5
+4ef 2e6f2 f - 4efl2 e2e6f22f4 - 4e2fle 2e6 f5 - 19e2f2e 2e6f 3f 5
__864f46165]
+2f2 2
- 42f2e2 e2e f f ++e2f3e2
-8edLf
9elf 1 3 e24 f 7
4 ele 5 fl~ + efIe2e 4 f 3 f 5 -4elf
4 2 6
-4ef e5 f2 f 6 + e4fe 2e4f 2f 6 - 2ef2e 3e4f 3f5 - ef2e 3e4f 2f6
-21e f2e-21(:,~~i
4e2fe
+ 4efe
2eaf
38 +le2
- 1 8f 92 f6 -9e2f2e2e
2 e6f 8
Ile2eaf8
f 4f 8 +
1 e2C6f8
2
-9e 2,,fl 2e8 f~2 f 4 - 21ef2fC2ee 8 f 2 f32 + 12eYf2e 8 f 3 f 5 + 12e12
f2e
3e 5f 8
11~
-~~~
~~-6elf
4 1e1f
2¢rer2
2
2
2 2
+2e'fe4f3f5
+ elfe4fe2f6
+ 2fe3e5f4 2
e22
efe2e6f
2
-e61 j1 e4 f 8 + e5f 2 e5f8 + e7f2e3f8 - 6e2f3e2f 7 - 6e3f2 2e7f42
2e
f8 + 6e2f]e8f42 e2e 8f-8 efe 9 f - 7elflegf33
-7efle3fg - 31efe 2e7f 4f 5 - 3elf4e2e7f 6 - 10efle2e7f3f6
+3e fe 2 e7f 7 - 8elfle 2 e 7 f 2f 7 - 3eif2e 2 e7f 8 - 11e2fe 8 f 4f 5
+6ef
-11el 2 fle 8f 3f 6 - 3ef,1 e8 f 2 f 7 - 14elf14e9 f 2 f 4 - elfl4e3 e6 f 2 f 4
+17e fle 2 e7 f 2 f 4 - 4e1fle 6 f3 f6 - 3e fle 6f 2 f 7 - 3elf e3e 6 f42
-2elfl 4e3e6f3 - 7elfle3e6f4 f5 - 4e flae3e6f 6 - 4elfle3 e6f3 f 6
+4elf3e 3e6 f - 10elfle 3e6 f 2f 7 - elf2e3 e6f 8 + 11e3fle7f 4 f5
+4el'fle 7f 3f6
-el j
e2e5 f/
3el f el 7f 2f 7 + 17elf12e 7f42- elfl4e2e7 f32
-
e
2fee
5 f4 f5
+ 5e f4e2e
1
5 f 6 + 18ef
l
e2e5 f3 f 6
-5elf ae2e5f 7 + 12elfe2e 5f 2 f7 + 5elf2e22e
5f 8 - 8elf2e 4e5 f42
+23elfle
- 5elf4e 4 e 5 f 6 - 7elf l e4 e5f3 f 6 + 11el fj3e 4 e 5 f 7
lflf
f
-llefe
13ef 3 egf3 f 4
2 7 4e 5 f 8 - 5ef 1e 6 f4 f5
-46e lfl3e of22f3 + 33eif2e 9 f 2 f32- 12elfl2e9 f3 f5 + 13elfl3ef 2 f 5
-42elfle 9 f 2 f 3 f 4 + 15efl5egf 2 f 3 - 12efl2egf 2 f6 + 33ef egf22f4
+ellee
e 2e3 f8
3 f 4 f 5 -3elfle2e
3 f 3 f 6 + 5elfle2e 3 f 2 f 7 -7ef4 e5 f 4 f 5
-31el
-3e6fle 4f 2f 7 - 19elf3e3 e6f 2f 5 + 3elfe 3e6f22f4- 12e3fle7f 2f 3f 4
-16el fl3 e2 e7f 3 f 4 -5elfl3e 2 e 7 f22f3 + 9elfl2e 2 e7 f 2 f32
+13e f2e 2e7f 3f 5 - 14efj3e2e7 f2f5 + 21elfle 2e7 f2 f3 f 4
+elf'5e2e7 f2 f3 + llelfe 2 ee7f 2f 6 - 23elf e2e7 f22f4 + 26e fe 8 f 2f 3f 4
73
3 - 5ef l e7f2f5
-7efe
3 e6f5 - 3elfle3e6f23f
8 f23f
1823--3 + 4elfle
l
-3elf 1 6e2 e7 f 4 + 12elfle 2 e 7 f2f 5 + 3elfl5e2 e7 f 5 + 5elfle 2 e7 f23f3
ff 4
+5e2fle 8 f22f5 + e 3 e,fl
4 e 5 f2f5 + 18elf2e 4 e 5 f 2 f 6 -elf 1 e4 e5
2efl2e2e
5f 3f 5
+5elf3e 3 e6 f3 f4 + e l fe, 3 e6 f22f3+ 6efl2e 3 e6 f2 f32+
2
I~~~~~~~~~~~
-3ef ee 5f2f5- 3efl ee 5f 2f3 f4- 17elf2ee 5 f 2f6
+5elf3e 4 e5 f 3 f4- 3ef2 e 4 e 5 f2 f32- 12elf2e4 e5 f3 f 5 + elfle 4e5 f23f3
+5eifle
6 f22f5
4f1 e3f2 f7
+ 18elf l e3 e6f22f5 - 2e4f1 e f3 f6e
-14e4fle 2 e 4 f 9 + 13e3f1 e2 e5 f9 - 12e fle 3 e5 f 9 + 33e2fle2e4 f9
l e3 e4 f 9 - 46ef l ee 3 f 9
-42elfle 2e 3 e 4 f9 - 12e fle 2 e6 f 9 + 13e3f
+33e2 fle 2 e2f9 + 15e5fle 2 e 3 f9 - elfl e4 e 5 f22f5 + 20e3 e7 f l f 4 f 5
+5elf e2 e 5 f 7 - 4ele 3f4 e 6f 6 + 3eloff 3 f5 - lOelof2f - 4e4 f4 e6 flf 5
l2 -14ef2e 6f2 + 4ef 4e2f l f5
+4e4f2e le5 f2 - 4e4f4 ele5 f 6 + 9e4 f4e6f
5
-18e2 f 4 e l e5 f 6 - 8ef
4 e 6 flf 5
-8ee3f2e6f6 3lfe
6 f7
- 4e4 fl e6 f 5 + 9e~f 4 e2f 6 - 14ef 4 e2 f6
fe 4f3f5 + 2e2fe4f 2f6
f-2e
e2
6f2+
j~4 + 1Oe
-e2f2eef2
2fe6f3f5 + 6e2f,2e6f2f6
e6f2f32 + 8e2f e6f22f4
-4e2f
+4e 2 f 2 e8 flf 7 + 4e2 f 2 ele 7 f8 + 8e2 f 2 e8 f3f 5 + 6e2 f2esf 4 + e2 f22e8 f32
2e2fef4 ~+ 1Oef2ef6
e2f2e f --+ 22
~ 2
2e3f2e4ff
+e
3J4f64
3 4 - 22~ef2
'2 2 f2e32f
f7 -4ele4f 2 3 f7
e
-3elf
f
e
3e-f
±
3ef3
3 77f7
± Ilhe~~~~~~~~~e4ele
7 4
+2efaef 12ff7 +12e
3 e3 e4 f7
3ef
f + 3e4f e2f
+ 3e6f e~~~~
e~f ~
e4f
fae
3el
2
~
~
5
f
f e f7 - 9ele f e f7 - 2ele2f e f - 15e4e e f
+12e3f 3 e2e 3 f7
+
3
3
3 4 7
7
3 3 7
-3ele 2 3 3
2 3 6
2~~~~~~~
-9elef 3 e 5 f7- 24ee 2f 3 ef7 + 3ele2 f3 e3 e4 f7
7 -
3 e5 f 7
2 3 4 7
-7eef2e
7f
23
+ele 2 f3e 7 f 4 - 12e2 e 8 f22f3 + 24e 2 f 1e 8 f 3 f 4 + 2e 2 f e 8f 8
+5elfle 4 e5 f33- 21elflegf22f 5 - 7elfie3f 3 f 6 + 5e3f 2 e2 f 3 e5f5
3
-ele 2 f3 e7 f7 + 3ele2f32e5f4 - 3e2e2 f32e6 f4
22
+2e2fef
6
15ele 2 f 3 e6 f 2 f 5
2
+ 2e 2 f 2 e2f4 + 2e 2 f 2 e2f8
4
44
e f4
- 4e2f22e
4f 6 + e2f4e6f32+ 2e2fe 6f 5 - 3ele2fle 3 4
+15e2e2 f 3 e 6 flf6
2
- 12e 2 f 2 e1 e7f4
+
-15ele 2f 3 e3 e4flf6 + 12e2 f 2e 8 f5f 3 - lOe 2 f 2 e 8 fl4'f4 4e2 f 2 e8 f 3 f5
+4e 2 f 2 e3 e5 f 8 - 12e2 f 2 e4f 3 f5 - 12e 2 f 2 e4f f 7 -12e 2 f 2 e3 e5 f462f3 e22f2
4
e3
+IOC2
+1e0~f2 e6 f2 + 5ef 3 e7f2f5 - 5ef 3e 6 f 2 f 5 - e3 f3 ele6 f7 + 6ef 3e2f7
+1ef 3 elf7
-
7 f4
11e3f3e4 f7 + 6e3f32e7 f2 + 11e3f33e7 fl - 11e3f32e
2 e7 f3 f 4 - 15e3f4 f 2e 7 f4
-e 3 flf 2 e 7 f7 - 24e3f 2 f2 e7 f32+ 3e 3 flff52f,
fl
24e
-e2flfef,
-7e~flf 2 e2f 7 - 3e3 fl f 2e2e5 f7 - 3e3f 5 f2 e7 f3 + e flf 2e4 f7
-2e 3flf23e 7 f 3 + 12e3 fl e7 f 3 f 4 + 12e3 fle 7 f22f3 + 3e3 fe 7 f32
2
2 2 f 2 f4
15e2flf4f5
-5e2ff23f3-3e
3 f3e7f7 -4e 3 fj2f4e7 + 5e f
+ Oe52flf22f5
-5e2f3f2f5
+ 1Oe~flf3f6
- 5e2f3f3f4 - e3e3f2f3e4f5
-15ef2f3f5
+ Oef12f27
+5e2ff2ff32
555
2 2
1
0e2;2;
+lOe2f2f3f5-15e~ff2f6
-4e
+~e
f+f
10J1f3
e2~~~~~~~~~5J
e e f 22
f
5e
+
f
f
e
f
3ele
f
e
f
e3ef
±
f
e f f e 2f3
62e
~ ~ ~3
~~~
3 2 3 5 5
3+1
102,
2 3 2 5
2 3 7 2 5
e~flef5f
+3efe 4f7 + 18eef 3e4f 7 + efe 3fe 4f5 -2eef
3
3 1 2 5 7
2 f3 e4f5
-13eje2f2 f 3e5 f 5 + 9e2f_1flef5f 5 - e2e3f 1f3 e5 f4 + e2 e3 flf3e 5f2
-4e 2 e3 f+f 3 e5 f5 + 18e3 flf22e 7 f 4 + 15e3 ff 2 e2 e 5f 6
+15e 3 ff 2 ele 6 f6 + 5e 2 f 3e3 f 2 e5f5 +3 3e3 f 22e4 f 5 +Oe21e4 2 8 -Oee 4ff 8 + 12e3f e7 2 5 -3eff
+C~2f-e4
+ 182fe+ef7flf5 -
-15e 3 flf 2 ele 6 f3 f 4
3e3flf 2 e4 f 6
2 e4f3 f4
U3 f~'
74
ff
-5e 31f 4 e 2 e5 f 6 - e3f3 e1f 2 f 5
+3e,3 f4e 7f6
-e3f32e
11e3 3 e7 7 + ee33 4 fl - 3e3 f3e7 f5
4 f 4 - 2lf22e8 f 6 + 2e'af2 e 6 f 8 + elf 2 e2f 8
~ 328f4 + 7elf22e
2 8f32
2 + e2fl3ef
-2ef 2ei7f~8 + 2elfie
e2fl el
3
3ff 2e5f 7
3-e 3
-9e 3 f3 ele 5 f 7
l f 4 + 17e 2 e3 f 2 f 3 e5 f 5
+5e2e 3 f2 f3 e5 fif 4 - 2e2e3f2 f3 f 5 + 5e2 e3 f 2 f 3e l e 4 f 5 - 2e2 e 3 f22f32e
5
3eafa~eflf6
1ef5
32f
3 eff
6 + ef3e
-13,e2e3flf22e5f 5 - 2e2e3f2f2e 5 f4 + e6f 3e4 f 7 + 4e2f 2ee 4f 8
-6e:'f 2 e2 e 6 f 8 - 12e f2 e8 f3 f5 - 13 f2ef 8 + 2e'f 2e4 f 8 - 2e5f2 e 5 f8
-2e!f2e3f8 + e f2e f4- ef~e4f6 + 6ef2e7f4 2-6e4f2
f8
-1Oele 2flO- 1Oee4fo + 1Oel7e
flO
+
1Oele
flO
1e2e
3
5
4 fIo + 1Oee4fIo
+10eof 4f6 - 1Oelof32
f 4 + lOeloff3 + lOeloflfg + 10elof3 f 7 + 35elof6f2
-5O0e1,of4f23+ 1Oeo2/
4
-10el~~~ ~~~~o/l24
- Oelof2f ± 15of
+10e3e
7 flO
-
+ lOeofl 5
+
5
ofx+I25e
4
+ 1OelOflf7f
1Oe1of8f 2 - 1Oelof4f 6
4
2 52loflf2+ 5elefo
- 5eefo + 15e2e2flo
1Oe2e8 fO + 10e3e 7 fl0 + 1Oele3flo + 5e2e4 f52- 5ee 4f52
~2+ 5eff4
2f2f - ~
lg~-5e5f3
eff ~- 5eff4-5ef7
3f ~ f755
f
-5e f4f6
5eff
5~~
5%f
~ f7
~~~
2f146
+5ef 2f 2 +5eflf6
+60e(-loflf2f3f4 - 1e
2 2 2
e2 2 2
5ef2f
+ 5e5f1f
2ef9
+
2e52f3f7
2 e 8 flf] + ef22ef
6 + 2ef22e 5 f 6 - 2e4f22e6 f6
-2e,]f2e 7f 4 + 20ele 4 e5 f2 f8 + o10ee4 e5 f2 f 3f 5 - 10e4 e6 f2 f8 -8eff
3 e 6 f6
+lOeef 2 f42- 15elegf22f32- 1Oeef2f 4 - 10ele9 f3 f7 + 1Oelegf3f4
-10e, 1e9 f4 f 6 + 30elof2f 2 f 6 - 20el0 flf 3 f6 - 40elof3f 2 f 5 + 30eloflf22f 5
-6Oeof2f2f32 - 20eoflf 2 f 7 + 1OOeloflf22f3 - 20elof 2 f 3 f 5 -60elof5f 2 f3
-40elof3f 3 f 4 + 50elof14 f2f4 - 40eoflf23f 3 - 20eloflf 4 f 5 - 60eof 2f22f4
-15e 3 e 7 flf ± 1Oe3 e7 f 2 f42- 1Oe3e 7 f2f 4 - 10e3 e 7 f fl 9 - 10e3 e7 f 4 f 6 - 2e2flo
2
2
f22 f e f -3ef C
-4eif2e
2 ef 6 - 9e3
7 f 5 - 9 3ffflf22e
2 7 6
3 e4ff 5
e32J3e4f 2 f 5 + 13e3 fe 4 ff 6 3e3 ee 5f4 15e3 3 e2e5ff 6
+10e 2e3 e5 f 1 f4 f5 + 1OeIe8 f3 f 7 -1Oefe 8 f32f
4 + 1Oee8f 4f 6 -Oele
-9e 3,'f 3e7 f2 f5 - e3f 3e7f 2 f 5 - e3f 3 e7flf 6 - 19e3f 3ele6 f2f5
-e3 f3e 2e 5fl + e3 f3e2 e 5f 4 - 19e3f3 e2e5flf 6 + 3e3 f32e
1e6 f22
+13e 3f2e 1 e 6f 4 + 15e3 f 3 ele 6 flf
-5ele
2 e 3 e 4 f52 + ef52 + 20ele
5
7 f4 f6
+ 20e2 e8 f l f 4 f 5 + 20e2 e8 flf 3 f 6
e f f
+ lOe
3 6 2 8
4 e6fl2f 8 + 5e2e 2 ef52
~~~~~2
5
-15ele 4e5f5 - 15e2eiff - 5e~flf 2 f3 f 4 + ef25 + 5e~flo+ elof10
-2elof2 + 5elof52 - O1el o flo + 1Oe2f52- 1Oe3e4 f 3 f 7 - 1Oe2e4 flf 9
+1Oe2e 4 flf
+20e4e6flf
-5ee
3
e
-5ee2e 3 f
+ e3 fl4e7 f23 + 3e 3 f6e 7 f4 - 15e 2 e3 e5 f52 + 20ele 4 e5 f 3 f 7
2 f7 + 20e4e6f 2f3 f 5 + el°f1 o + 20ele2 e7f 4 f 6 + 5e2ef
- 5ee 2e 5 f52 + Oe1e 2e 7 2 + 5e2ee 4 f2 - 15e 2 e6 f
9
loee
+ 5f
1Oelee
+ 6f
+20e 2 e 3 e5 f 4 f 6 + 60ele 2 e3 e 4 fIo
75
+2Oel
5f
20e2e3e5flf 9
Table of Pm,2 for m > 3
4
2
e8
el2e
e
e
±
e
ele
+
e
6 + ele7 + 2e 2 e6 5
3
5
2 5
m Pm,2
e1 e 3 e 4 - 3ele
5
e4e6 + e2e4 + 3ele 2e 7 + 3ele 3 e 6 - 4e2e2e6 - 2ele 4 e5 - 2e2e3e5
±e2e3e5 + e 1 o- e3 e7 + 2e2 - ele7 - 2e4e6 + 22e 6 + ee 8 - ele 9 - 2e2e8
6
elell+ 3ee 6 - 3ele2 e3 e6 + e 5 e7 - e4e8 + ele
2 e4e5
- e1 2 + e4 - e2
0
1 elo
2 q-e 2
3e3 e4 e5 - ele2
e8
+
+2e2 e12 ±
ee3e7
2
232
+e3
+±le 3 e 6 - ele 4 e 6 + 3ele 5 e6 - 5ele 2 e 7 + 5le2e 7 - 22e 8 + 4e e3 e7
ee 7 ~2 + e1e
2eI
e+ 93 e3~e9
-3e 2e 3 e 7 - 3ele 4e 7 + 4e2e2 e8 - 3ele 3 e8 - 3ele 2 e9
7
7le2
2
ele4e5
+3e
+ 3
5
e 5e7 - 4e 3 e 4 e7
22
e2
e2 + e2 2
+ 4e
3
ee 5 e 6
+ 5ele 3e 8
e7
ele 4 e7 + 4ele3e7 +2 e2e 3 e7 + e4e 3
4ele 6 e7 - 4e2 e5 e7 + el4 + 3e 2 e3 e9 - 4e2le3 e9
-
+9e2e2e8 - 6ele 2 e8-
2e
6
ee
3e 5 e6 -
+
-5ele2e9 + 5e3e2e9 + 3ele5 e8 + 6e2e4e8 - 5e+e4e8 ± 3ele2ell + 3ele3e10
-4eIe
-2ele
-9ee
8
3- 2e2e8 + e6e8
e 2 -2e 2 e1 2 el+
ele4e9
3e
2
el2e2 e 4e6 - ele 3e4 e6 + ele 2 e5 e6 - 4e2le2 e3 e7 + 2ele 2e 4 e7
o - 2e4e10 - el-e - 2e6 e8 2 e 3 e8 + ee 6 - e 5e 9 + e 1o + 2ee 1
2 e 10
22 +
3e 5 +
2
2e2e3 e5 e6 - e2e3 e 5e6 + 3e2e8 -
ele2e 5e6 -
e4e5 e7 -
e9
3e4e5 e7
+2ele 6 e 7 + elae2 e4e7 - 2e2el 2 + ele 3 e 5e 7 - 3ele 2 e6 e7 - 6e2e 4elo0
-3ele5 el0 - 5ele 2 e11 ± 5ele2ell - 3e2 e 3 e11 - 3ele 4ell + 4e2e2e1 22
-3ele 3 e12 - 3ele2el3 - 17e e2 e3 e9 + 12ele 2 e4e9-3ele2e 4 e7- e ee4e7
+5
3
3ee 2 e 5 e7 + 2e2 e1 4 + elel 5 + ele 4 e 5 e6 - e2le2e6
7
e16 ++ e22e+
e 9 - -e1
e6
e
4
e1 - +
e - 2eee
-5e
5elee
+
e
ele
+
2 e3e8
e2e
8
e
e
+
3 8
14
3 13
+2e 4 e12 - 3e 3 e6 e 7 + 2e 2e7
3e9
a2
- 2eg
7e5
2e6
2 2e2e
e~~e2
+
2
e3
14ele
e2 e9
ele4 6 6 e
e e + 5ele2e e
+3ele
3
-5e3e2
2
3 8
3 8
e2
3 4
-7e l e2e 9 + 6ele 3e9 + 5e2e3e9 +
2eee
+e 5 e1 1 + e7e9 - ee1 --2o
-3e 2 e 5 e 9 - 3ele 6 e9 + 6e4le2elo +2e3e 5 e8 + 5e 3 e5 e8 - 3e2e6e8 +
9
e
2
~~~~2
2
3ele3e 9 + e2e6 + 9ele 2 e3 e,10 + 4ee 3 ell
- 6e e4e9 - 3e3 e 4 e9 + 4e e5 e9
9ee2elo - 5e3e3el0o+ 2e 6el 0 - e4le4e8
2e2 e6e 8 + 5ele 7 e8 +-3e2e2 e4e8 - 8ele 3e 4 e8
-4ele 2 e5 e 8 + 5e2e4 e1 o + 2e3el0 + e3 e3
2
eo
7 + ele 2 e4e 5 e6 + 3ele 5 e6- 8ee 2 1
1 2 -2 2e2e5 e7 - 3 3e5e7-2ee5e
16e2e2elo + 7e5le3 eo + 7e2e2elo - 3e2e2elo + 3e 2 e3 el3
2eeee-ee
e2e
e2e2
2
~~4
6
-6e 2 e 7e9 + e61e2+ 3e 5e 6 e7 + ele7+ 2ee7 - 2e 4 e7 - 3e3e4e5e6 - e5e8
+20e4e2elo
-
22e3
2
2 eeese
27e
- 5 el
6e2e 6e 10 - 27ee
0 14
2 e3+e 2e2e
2 e3 e 12
±7ele~e
e1 2 + 4e2 + 6ee
6eeeeio
0le+2e4ell-9ee
3e3e
3 64e-ele
2 e4+
6 +
17 +
2 e4 eo - 8ele3 e4 e 0 -2e2e 3 e9 - 12ele 2 e3e + 4ele 2 ee9 - 2eee
+19eiee 3 eo + 21ee
e3el5 + e2e - 2e2el6 + e6e3e9
4
e9 - 1lOe2e
3 e 4eo
+e1eee8
5eale
+6~~~~~~~~~~e2--
2
~~~~~~~~~3
3
+6e 2 e3 e4 e9 - 7ee 2 e5e9 + 9ele3 e5 e9 + 6ele 2e 6 e9 + 3e3e9 + ele4e7
e2
+3e 3 e e 7 + el8 - 2e8 el 0 - 6e41e2 el2 + 9eeel2
1
2
ee 6e8
36e9 -ele 4 e9 + 5ele4e9 + 2ele5 e9 + 4e2e5 e9 -3eIe 6 e9 -6e 3 e6 e9
eCe
+5e2e 7 e9 -6ee 8 e9 + 2e4el 0 - 2e2e3 e5 8 - e5 el3 + ele 14 + 3ele 7 e10
- 5e2e3ell - 3ele2e
+7e5e2e - 14e3e2ell + 7le3el - 6e4e3 e11
76
+6e;e 4 e1 1 + 3e 3e 4 e11 - 4el2e5 e1 1 + 3e 2 e 5 e11l + 3ele 6 e11
22e 4 e8 - 3e3e 4 e 8
-4e 2 e64e8 - ele 5 e8 - 8e2e 4e1 0 + 4e1 e 2 e3 e5 e 7 - e62ee 6 - e2e5e6
+ee+ e2e
+`.:
- 22e
~,e-e~
C4 e66 - 6e4e5e9 ±+3ele3e14 - 7ee 4 e1 0 ± e2 e3 e5 e8
ee
+
+2e1 e4 e5 e8 - ele 2e6 e8 - 3el e 3e6 e 8 + 5e l e2 e7 e8 - 3e1 e 2e 3 6
e64e
10
+el4e 2 e 4 e8 - 42ee 4 e 8 - ele 3 e 4 e 8
4le 2 e 5 8 - 3e1 26e
5e 8
-e1el + e3 e5 - 2e2e6 e8 + 6e 4 e6 e8 - 3ele7 e 8 + 3 3 e7e 8 + 2ele8 - e 2 8
-2eel4 - el3e1 5 - 2ee 12 - 12ele2e 5e10 - 6ee 2e 3e9 + 9e2e 3 e9 + e8lel0
-e711 - 5e 6 e 10 + 3e 3 e5 e10 + 4e 2 e 4 e 5 e 7 - ee162 e6 e 7 + e1 e2e6 e 7
.5ee 3 e6 e7 - 3e2e3e6 7 - 5ee 4 e6 e7 - 32e 2e6 - 3e1e2 e42e
7
e12e22e
5e 7 - 2ele 3 e5 7 - ee 1 3 - 5ee 4 e 12 + 3ele 5e1 2 + 5el3e2 e13
-5ee22e13 - 4e 3 e13 + 3el e4e1 3 - 4e62 e14 + 3ele2e15
10
-el4,e2
8
57
± 46e2- 3e 5e 7e8 + 5e2e 5 e1 1
4 +8e9
2e3e5 e9
-3e;e2e7e8 - e6e5e9 - 7e e 3 e 4 e 9 - 4e l e1e 4e 9 + 3ee 5 e9 + 2elee 7e8
+e5e
6e9
-
3e 4 e 7 e9 + e15e 2 e 4 e 9 -
5e2e
,
4 e9
2e 2e 3e7 e 8 -
34e 7 e 9
-3e6'e 3e7 e 8 + 4ele 4 e 7 e 8 + 3 2e - 32e - 3e 3 e8 e 9 - 4e 4e5e 6 - 5e 2 e1 5
+5e1 e2e15 + 42e 3 e15 - 3e2 e3e1 5 - 3ele4e1 5 + 4e2 2e 16 - 3el e2 e17 - 3e
+ e7
+2eie2e82 + e 3 e52e6 +
ee 3e10 + ee 2e3 e5e9 + 2ele3ee28
+19le 3e 2 e4 e 0 + 4e 3 e42e
9 - 22e 16 - 3ele3e1 6 + 7e1e e
1 0 + 16e2e3e 1 2
5e14
-76 e 3e1 2 - 7e3e 12 + 7ele3e1 0 + 4e 3 e4 e5 e8 - 4e2 e6 e8 + 3e2e 6 e8
22
4
e
2
C
+7e2e
3 e1 o - 2~~~~~~~~~~-1
14le3 e4 e9 + 4ele 4 e 5e 9 + 3e -2e4 e 5ee9
9 - 2ele 2 e5e6 - --e6le
4e10
+8e l e2e5 el0 + 14e2le 35 e 1e - 13e2 e3 e5 el - 13e1 e4 e 5e1 0 -9e7e 2 e1 1 + 2e4 e 16
+e54 - 2e 2 ee 1 0 + 6e6e, 1 0 - 7e 4e32e1 0 - e2Le5 e6 e 7 + 5e 2e 5 e6 e 7
3 2e3e1 2
+7e4 4 e41 2 - 6e3e5 e1 2 - 3e 3e 5e1 2 + 5e2e 6 e1 2 - 3ele 7 e61 2 2e 5 e10 + 8e26e4e1 2
- 24 6 e10
62 1 0 - 3e 2 e7 e11 - 3ele 8 e11
8e6 2e 2 - 2046e2 1 2
+4ee7e1 1 - 5ee 2ee 9 + 4e4e6 e10 + 2 8e12 + 4e3e7 10 - 52 8 e6
10 - 4e4e8
-e6 1 4 - 7ele2e3 e 10 21 2 ee
e9e1
4e 2 e8e1 0 - 6e3 4 el3 - 812
_2
-e 12e3 e6 e 9 3e 2 e3 e6 e9 + 2e l e4 e6 e 9 + 7ele 9 e1 0 - 6 e2e2e
1e 2 e 5 e 9 + e 5 e1 5 + 2e 6 e1 4
-4e2
+2e
e2 - 3e26e4e
- 39e4e 2 e 3 e11
45e2e2 e 3 e 1
27e,5e2el1 1
12ele 2 e 4 e1 3
-1317- e4 e12
7e3 e7 eI - 6e 2e6 e1 2 - 17ele 2 e 3e1 3 + 3e 3 e 5 e6 -21ele
2 e 4 e12
+8e 1 le3 e 4 e1 2 + 12ele 2 5e 12 + 3e3e 3 6 8 - 3e2e 4 e6 e 8 - 7e5e 2 e 3 e 10
6
e24 65~
+14ee2e 3 elo + e,2e2 e4 e5 7 - 30e3eell+12e
25ele2e4 ell - 17ee 3 e4 el
2 e 3 e 4 e11 - 20le
2 e 5 e 11 + 9e l e 3 e 5 e1 1 - 3e6 3 e5 e 9 - 3e 1e5e 9
+4e2e5e7 - 2ee 12 + 6e 2e3e5e8 + 6ee 5e6e8 - 4lee 7 + e3Ce
+9ele4e 1 1 - 7e5e 2 el 3
14e3e2e 13 - 7ele3e 13 + 6e4e 3 e1 3 + 5e 2 e3e13
d-26~1 - ele2e7
2 2 2 + 2ee2136
1 ele
-.
6361
+3ele3el3
e3 e7-e26462
2 e4 e7 + e7e5 e7 + e3 e17
~7e16267 + --3e
e2e3 e6 e 7 + 5e4e 2 e 4 1e 0-5
2ee4 e1 0 - 7e 1e 2e8 e9 + 8e6e 3 e1 1
+12e:3eIel1 - 7ee 3 e11 - 8e5e 4 e + 5e,4e 2 e 5 e 9 - e1 e3e 4 e 5 e 7 - 4e2e7 e9
+ele1 5 + e2e4 e6 - 3le 2 e 6 e9 + e1e2e6 e 9 -e
262 18 - 4ee 2 e4e 8
18
-5ele
2 e 3e 6e 8
- e3e 5e 7 -
3e 3 e5e7 2--
3e 3e 4 el3 + 42e
5 e1 3 -
3e 2 e5e1
3
~~~~~~6461 e
-3e(e-6 e 13
6e'e 2 e14 - 9ele2eM - 5ele 3 e1 4 5e646 14 - 6e 2 e4 e1 4 + e17e
13
9ee 2 e7 e9 - ele 3 e7 e 9 - 3e 5e6 e 9 e7 e13 12ele 2e6 e11 + ele2 e4 e 5 e8
+4ele3 e~e8 + e3 e2 ee - 3ee 3ee - 2e4eee + 3C2e
+4e('C~e8
2
_
12058
2
e
165568 + 326558
77
-7ele2e 5 e8
ele 4 e
2eee
8
4e
ell + e9 ell + 2ele2 e4 e6 e7
+2e4e26 + 2e e - 2e2e6 + 9ele 2 e3 e1 4- 2ee 3e 4e + 9e2e2 e3 e4e9
elO
e4e1 + 7e4e5ell - 12e3e 3 e4elO - 9ee 2 e5~~~~~~~~~~~~~~~~~~2
.±elel9- 151e2e 2 ell + 32e3e 2-3
e2e14 - 4e2e e8 + 5ele2e4e9 - ele2e3e7
e7 2ee7
-e 2 0 -- 2c2e4 5 + 2ee
-e6elo
-8ele 3 e 6el0o- 8ele2 e7elO- e4le16
-6e3e 6 ell - 3e4 e5ell + 5ee2
+27e3le2 e 3 el2 - 19ele2e3 e2 + ele2e6 e7 - e2e3 e6 e7 + 5le32e6 e7
±ee 4 e 6 e7- 5e 3 e4 e 6e7 - 2e2e 6 e8 - 3e3e6 ell
78
Table of Pm,3 for m > 3
m
4
Pm,3
-el 1 + 3e e6 - 3e l e 2 e 3 e6 - e 5e 7 + e2e6 + ele 2 e4 e5 + el2 + e43
+elelo - e2el0 - e2e4e6+ 3e4 e8 - ee 9 - 3e3 e4e 5 e1 25 + e2e25
±ele 4 e6 + ele 5 e6 - eCe2e7 - e2e8 + 2ele 3 e7 + e 2 e3 e 7 - 2ele 4 e7
+e2e 2 e8 - 3ele 3 e8 + 2eze2 e9
2ele2elo - 4e2e3e10 - 3ele4elo
0 - 3ee2 ell + ele3 ell + 2ele2e12
-3ele 3 e 4 e7 + 5ele 2e5 e7 + e2 e42e
e 6e 8 + 2e3e2
5 + e4ell +
- e3e 9 + 2eje 3 e5 e 6 - 4ele 2 e6 - 2ele2 e4 e8 ± ee 3 e8 + 3ele2e6
-2e1 e 4 e5 + 3e2e5 e6 + ele2e 9 - 2e3le
3 e9 + 2e e4 e9 + 3e2 e4 e9
-4ele
5e 9
-2e 2e3 e5
-e3e2 -
32
+
- ele 2elo0 + 3e 3 e6
- 4e4e 5 e 6 ++ C2
e 2 el - e 2el 3
e+15 + e~2
ele 3 e2
e5
ee 2e3e9 + e2e7 - 4ee2e3e7 + 3e 2 e 4e7 + e4e7
e3e4 e8 - e7 e8 e2e11 + ele2e4 e6 - 2e2e3 e4 e6 - e 2e3 e 4e 6
2
+4eej0 + 2e2e2e
- e4e, + 3e2 eeo + el+e
7 + 4e ee7
6
e2e6e7
-2el>e6e7 - 4e3 e5 e 7 - 4e2e4 e7 - ele14 - elee 8 - 2e 2 ee5 e 6
328 + 5e3 e4 e8 + 3ele23
-2ece 5e 7 - 5eea2
5e8 - 3e2 e5 e8 + 3ele 2e 3 e8 + 2e,5
3
3
3_
~~~2
2e2
eleaese
e3 e5 + ele5
e
e
e
e
22e
e ee - ee_
eee
4
3
5
6
5 e6
3
~
~
2 ~2
2
e23 5 6
2e
2+
C3~elee 6 82+e2elo
4eee
-
-3e e4e5e6 + 4e2e6e,10 eee 6 + 4e 4e - 8e2e4e6- 4ee5e6
-2ele2e 3 e62± e3 3e9 - 2ee 9 - 2ele3e~+ ee 0 4e2ee 6
+3e]e 3e6 e 8 - 5ele 2 e7 e8 + e8
e2e8
e2e5 e7 - 5ee 4 5e7 5e5 e6e7
422
~
2
2
-2e e 7 e 2 e7
ele 2 e 5e1 0 - 3ele 2 e 3e11 -ele 2e4 el l + 2e26e
+4eje~ea
e+2 2
2e2e2
a3
e2
leC
+4ece 2 e3 e5 e 7
3 e6 + 6e 6 el1 2 - _ ee2e
q 6
e6
le 2e 4 e 7
-e_22e 3e 4e 7 + 8ele2e3e12
+2e,;e
4
6e 8
52C
e3e4e11 - el
5e
1 1 3 + 52ee
- e 4 e6e 8 + e 3 e 7e8
2
3
ee
8
8
+ ee4 6e8
± e2 e + 5e2e
-
+5eleel
3e 2e 3 el 0
+e41 e4 el0 + 2e2e4 el0 - 3ee5e 10
6e3 e5 el0
e12e1 6 - e2 e1 6 - elel7
+5ee2e 4 e 7 - 3ele 2e2e 7 + 4e2ee 4 e8 - 3eae 3 e4e 8 - 3ee 2 e5 e8
-3e e 2e5 + 2elee 1 1 + 4e2e 5 e1 1 + 2e 2e 5ell - 5ele 6 e1 1 e4Le
2 el2
-3e'e2e1 2 - 3ele3e1 2 3ee 4 e2 - 3e2e4 1e2 - 5ele5e 2 - 2ele3e4e6
-e3e 1 5 + ee 1 2 ee2 e3 ele
4 e 7 + ele 3 e4 e - e24el
0 + e18 + e4e5 e 9
-e 7 ell + 6ele 2e 4e 5e 6 - 2e3 e4e7 + e2e14 - 3e2e 2e1 4 - e4e14 - eel0
+3e3e 7e8 + ele 8 e9 + eeel 0 3ege9 - 5ele3e
8 - 5e~e4 e8
I
12
332 e4e
23 e2ea
2ee3
-5e 2e 6eI 0 + 2ele7 e1 0 - ele2ell
+ 2ele2e1 + 2ee 3el -2e2e
3 el
+ee 8 + 7e2 e3 e6 e7 + 4ele 4 e6e 7 + 5e]e 2e 2 e2e5 e 7 - 3e3e5 e 7
+4elee 7 - 2ee4e 9 + 2e4e 5e 9 - 2e2e 5 e - 3e3le6 e9 - 3e3 e 6e9
-2ele 7 e9 + e2 e7 e9 - 3ele2 e4 e9 + e2 e3e 13 - 3e3e12 - 5ee2e 3e 8
+5e12e2e e8 + 3ele2e 7 - e 5 e 13
e4
14
7ele
2 e 6 e9
-4ee
3e 5 e9
+4e13e2 e 3 - 3ee2e 13 - 2e2e3 e13 + 2ele4 e 13 ee 3el4 2ele 2e15
+e4 e 2 + 2ee 6 2ee2e 4e 9 + 8e2e3 e4e9 + e2 e3 e4e 9 - 2e2e2e 5e 9
-3ele 2 e 3e4 e8 - 2e3e4 ell - l 1 ee 9 + 2e2e4e 8
e2e 8 + 6e2e2e8
e2C2
7
ee2eq q+e9
+3e63 + 4ee2e
e
e 2 e3 e 5e6 - 3ele 2e3 el0
3 l9 - 7ee2ee
2
±2ele22eel 0 ee2 2 e 4e10 - 9ele 3e 4e10 + 3e132e2 e6 e 7 - 6le2e 6 e 7
79
-5e
e 3 e6 e 7 + 6elee6 5 e 8 + 6e
3 5 e8 -
9e 2 e3 e5 e8
-
2e 1e 4e 5 e8
-4e2e 2e6 e 8
7
3
2
3
33
-35e2e51e3e1 3 - 3612el 3
- 5e3 e2 -2e2e 3e12 + 2elee12
-ee62e12
+ele 8 el2 + 5e2e 7 el12+ 5e2 5 12 +1365e 1 2 - 2e1 e, 12 - eie4e12
- 43e 5 e613 + 2ee 4e 13 - e2e62e6
13
+e 2 e 6 e13 + 5e21e
6 e13 + e3 e5 e613
433641636e
2 2
e7
e
+ 5e7 + 4e3e2 1 4 - 9e 2 5
-4e13e 1 3 + 3e1e 3 e1 4 - 3ejee4
4e3le
4
+
4e8
-5e 1 e 7 e1 3 -e 2e 19 + 6e 1 e32e1
2 e4e8 + 5e26e3 e1 4 -3e3e
-6e 6 e 7 e 8 - 5e2 e5e614 + 5e26e5 e1 4 - 9ee 6e2 - 6e 3 e 4e 14 - 33e 4e 14
5 + 2e l e2e3 e 4 e 9
-3e e 4 e15 + 6e2e2e25 + 2ee 2 e3 e 5 e 8 - ee361
-5e1e 2 e15 + e2le5e9 + 5e 2 e5e9 + 4e2e6 e 9 - 2e5e7 e9 + 6e1 e4e8
-3
3e2
---5e,67eM~~1
31'
--e461 21 1212
e2e16
ee 0 - 3ee1e
6 +4e0
+3e
9- e6
1e4e16 + 8e2e4e
15 +2e
2e4e6
32e2
3e4162 e3 1e 2 - 4e1 6e
2 e 3 e 15 - 32e 2 e 17
+7ele2 e3e 4e 8 + 2ee 2 e18
+e42e5 e8 + 7e1e2 e 3 e6 8 - 5ele2e 3 12 - 5ce 3e36 e 9 + ee 4 e6 e 9
17 + ee 13 -6e 5e7e9
-6e 2e4 6 e9- 2e3eeo + 5e3ee21 0 + e4ie
-44le 8 e9 - 42e 8e9 + e1e9 - 6e4e5 e8 + lele 2 e3el12 + 8ee3e5e
+3e 2 e3 e5 ell -4ee 4 e5 e11 - 4e2e2 e6 e11l - 4ele 3e6 e61 1 + 96162 e7e 11
e3e2
+ 3el263611
+3e6e5e8- 263e5e8 + 162e3e4e5e6--412631ll
33
15e,126467 + 6e2e3e9-9
31 6
+5c6e2e312 l-3e62
2
4
I12e7e8 + 8el3e367e8- ele356-4e1
-4 le2e7e8 +
3651
26260
34~266910
e2 + e3e465e9
_11-1 26364C8
-4e2e-5e0
+ 3e-162e69
--12clC23el4
8
+ ,le3e4e2
9-12le32e36e
-7l 5110
+7ele 2e 4 e1 4 - 4e2e 3e 4e61 2 + e2 e17 + ele 3 e1 7 - 10ele 2 4e 5e69
621 - 5 2e 3e 4 6-4ee2e
3 e4e8
+4e 2 e4e5e8 -e 2 e4e5
4 223
_C5
-e65e2e14 + el62e13 + 7e2e4 e5 e10 6 2 4 65 610 - 616e2 e6 e610
12231
836462
2
2 e261623
e~~e
6e 2 e3e6e08e6e2 e6e10+ 613610
81261
16e2 e3e6
10
e10
- 2e1 e4
~ 2 e7,7l
4 e6 e120 - 11ee
4
6261
~~~3
2
- 2e36ee
6e 2 e7 e12 5 e67-4e 2 e4 e1 3
-4e e2 e 5 e12 + 11ee 2 e 7 e9 - 12e1e2e 7 e - 11e 3e 7 e 9 + 12,e2 e3 e7 e69
436~1
e2e3el + 4e
2 + 7ee3e
2 eel 0
1 0
+6ele 4e 7 e9 + 11lee+2 ejje2
8
8 e 9 + 6e1 6e6e
-2ele
3e16 -
3~~~~
2e2e
3 e 5 e7
e2
+2ee 3e 4e6 e7 + 4e1 e 2 365 7 - 2e2e 3e4 ell + 15ele2e3 e 5 e10
+ 5e2leell
+5e3e
1 44 - 4e2 eell+ - 2ele5 e11 - llele3 e 5 el2
3
+5 4 ei1 + 6e32e4ell
-5e126e 5e9 + 161e32e5e + 66le 3 e5 e9 -13e2e
3 e 5e 9
+ 6e1e3e 5 e9
±
22 - ee ee+6eee-3ee0+
+5626667
+ 10ele e e10
+ e e3 e5 e + 4ele e6eleee
-3e e e e +3e61e6e7
1 2 5
3
6
6
2
4
6
3 7
6
e + 6e3e 4 e6 e8 + 2e 1 e 2e 5e 13
-5ee 2 e 8 e10 + 7e2e 3 e6 e 8 - 8ee2e
~~~~~~~~~~~
3 6 8
2 ~
2eCe7 e8 + 2ee2el - 4e2ee
- 6e166 614 + e2e1 9 + ee+6e 614
7
-3ele 3 el - 8C13 + 2ele 2 e6el62- e2e3 26 e7 + e42e 13 + 19e 1 62e 3 e 6 e9
+2e 2e e6 e 7 + 3e4e 4 e6 e7 -9e2e 4 e6 e7 - 3ele 5e6 e 7 -2e 3 e 5 e6 e 7
+5e e2e4 e6-
36ee
6e 9 -
ee5 + 6e3e7 e8
-6ee
4 e8 -
17
5 e8
+e 4 e 5 e1 2 + e 2 e 9e6
1 1 -6e 4 e 5 e -6eje 2 e3 -e2e3e
1 0 + e2e24e
ee
+ee 4 -2 e 2e4 5 eee 3e5e7 - 5l61-6e 1e 3 9
2
e5e6
- ee5e6
e61646e
+3e5le82- 4eee 4 e7e8 ± 4e2e4e7e8
2" 4 2+2 C2
52 e~~~~~~~~2
+e2e
2 e4 e6e8- 1013 e 4 e-5ee
5e616264668
d-613e646-
ee -6 e 0
- -6516661
80
3e4e7 e
6e5 e6 e 0 ,1 364167e10
+ 66,56610
1,
+5e26e
e - 5e e e + 3e3e8 el1 0 + e3 e8 e10 - 2e2e 9 el 0 - 3e
+61:1o 7 1 0 e1 74 7 1 0
+el620 + 324C7 + 3 624 e7+ 3e364e 7 -3e 2 e43 7 - 6 5 e
1 6
2
~4
C2C2
6
+12e6e 2e23e 4e10 + el15e
+
4e2e5
6e24e
e
+
e6e
7
7
4 9
1 5
2 3 2e
4 7
-3e 3 e 12 + 5ee26 4 e9 - ee 2 e4 e5 e 7 - 7 e2e3 e 5 e9 + 3ee 3 e1 5
+6
eee + 3ee3 621- 3e e4 e5 e 6 - 3 ee 5 e6 + 5e2 e2e 5 e6
----C3 13(:5 ,26
q
-102e 5e6 e8 - 2e 2e 5e6 e8 + 8e 3 e4 e6 e 8 + 5e3e2 e3 e1 3 + e5e8
++~3e -11e~~~~262
+e3-251 22
-9e;3e 2 e3 e4 e 9 + 5e 4 e 8 e 9 +
+3e'6e2 e4 e11
5ee2e 4 e11
-
1eIe3e
-8eIe
e 2 3e 4 e9 + 2ele 2 e4e 9
4 e9 -
3 e 4 e6
1
-4e1 2e15e - 33e5e7 + 52 3e4e5e7
+4e'!e5e - 2 2ee5 8e - e 2 e6 5 8 -4e
+ 43e 2 e5 e1 1
-2
4 e5 8
9 3e4e7
- 2e2e
6 e11
+9e 3e6 e6
1 2 + e 2 e 3 e1 6 + 34e9 + 5e 4e2e 7 + 2ee2e 3 eI3 + ee6
3 ~ ~~~~~~~~~
26361
2
1
3
+3e3 e -ee2 - 9elee 5e7 - 4e 2 2e 7 - 61e3e2e
7 + 7e1e3e4e5e8
6
4~~1
5
2 2 e2 e2
e2 + e4
2
2
±3el3 e~e 1 -ele2e~e~e
4 e 103ee 15
0 e
1 3 1
2 e3 2
±el~e~e~e~e
+
5e~e~e~e
- ee 2e~ee10 + 9elee~e
7
0
+7e:t2C8
+
76368
-2C2C382
3
IC2e3
7 ll~~
+e2413e4e5e7- e18 - 5ee2e4e07-4eIC264eIO
+ 03le2e4eIO
+3~1e341o -26236461o -1561326e461o-2
6~4613
IC~~~~~~~~
2410
~~~~3616
-
+ele
2e4e5e77 q+ 5el2e3e3eO
d-124
21
-5l3e2eC2e
3
8
- 9ele2C2elo
2
+3ee2e 6 e 8 - 7e1e2e6 e 8 - 4ele 3 e6 e 8 + ee 9 + 62e11
3 1 e3 e 4 e13
-23e 1 le 2 e3 e 7e8 - 6e1e 3e8 e 9 - e1 2 9 + 6e 6 + 61e24e4e 8
-2ee264 2 6 2+ 3e4e 6 e1 1 + e 4 e6 e1 1 - 43e 7 e11 - 66367611 - 262e 8 e1 1
+2e=,e 8el + ee 9 ell- 2ejee7 + 32e 3e7 + 3ee3e7 - 5e e4 e7
-6].e4 e3 e9 - 2e3e6 e7 + 2e4e 6 e7 e 3 e4 e8 + 2e32e 4e 8
+e 1e 5 e1 5 - 4e3e6 e 12 + 5eje 2 e5 e6 e 7 + 3e 2 e3e2e 8
-3ee9ell
- 5e 3e 7 eI 4 + 1e 4 e 6 e
- 6e5 e 8 el
+ 62e
62e14 + 44e 6 e1 4
~~~~~~~2
-3e4'-e82 6268614 - 2e2e9e11+ 6e864 + 4ele2e- 5 25
+e2 621 -_ 2e62 + e361061 + 3e3e 0 e7 + ele9e14 - 7e--4e10
+646396
+ 43e -6162666
6e3ee-el~
23e4el
+ 13
7-- 173
612636468e~~
1
-6e18 + 3ee3e15 + 4ele2e5e62 - 64e6-5e2
+ 8e2 e2e4e6eIo
+26e3e1 + e62e
2
- 3ele4e1 - 5e632612- 42e3CIO
-9623676122
2e25el5
6e71 + 163ee
+ 1 + 22e
4 5 7618615
1 4e
-7e3;e3e_
~~2
2
+ 626715 - 56l6661 + 12e1-e2e3e5elo ± 6l6261
-46e263e8 66613
-
2 7e3
24713 - 213
+2e2'e2e5e9
+ 5el 6 3e
d-22
3
32 5 e9 + 14e3e2e8e1
-
3e2e2e2
3017
+ 4e4e5e15
± 6ee12
6
-8ee2eee9
- 5e 4c2e16 - 12ele2811
- 9e~e4e6e8
-7ele3 68-7ee
6e3ee 5 e10
5 e 2- e34e34-610619-eO4+6e462e16
+4ele3e
+6e42
e3e4e1 2 ± 15e612636 13
- 2e2e3e13 - 66321 + 2le2e21 + 3e65ele2e3e6
156 22e67
-3e
27e3e5e16
+ 6e2e361+ 29ele2e3e2e
+6e2O + ele3e20
-
1OC6I2e3e4s5e8
-
3e1e3e6
± 361626615
- 3e1e2e3e7e8
+2e 51 - 4e3e2el + 6e3e12ell
- 6666ele2eell
+ 22e1e3e2el
+3e2e3eel
- 14e3e5e2 - 14e2 e3e8e1~4- 627e2e6e14
28
81
2 3a
2
2
7e3e2 e4 e7 e8
ele
2 e4ee7 - eFe 6 e 7 + 5e2e e6 e7 +
-le
~~~~3
3
2
3
+ 3elee2 e e -18ee
-7e3e5 e10 + 6ele6el6 + 4e4e12
3 6 8
66e
6e7e9
+6e 2 e6e7 e 9 + 5e5e2 e8 e9 + 7ele 2 e4 e5 el2 + 7e2e 6e - 3e5e2 e7 elO
5e5ee9
~23
5
12
-6e 2 e3 e 5e6 e8 - 8e3e4 el4 + e2el4 + ele2e 3 e5 e8 + 5e e4 e6 e9
+3ele4e6 e9 + 8e3e4 e11 -3ele 3e 6 elO - 9e2ee 6 - lee 2 ee 6 e9
+6e2e3 e 7 e8 + 3e3lee 7 e 8 - 3ee 4 e7 e8 + lele 2e 4e5 e6
- 20elc~94-e8 - ee -4-5
5 e9 + 5e2 e4e5 e9 - 8ee 2 e4 e5 e9
-14e 2e626
4 e5 e9 -2Ole4e
- 17ele3 2el 0 + 2e6e5e13
-2e2e3e7 + e2e3e7 + 3e4e4e16
3
4
2 4
~ + 6ele2e
e~~~~e
3
3 ell + 12e2 e3e 8 ell - ele 2e6
+3e2e 5 el3 + 2e3 e4el3
~ -2e2e 5e17 -7e 2 e 6el6 -14ele 7 e22
~ 72el6 ~~~
+ ~2elee12
- 6ele
+4e2le2e 62~~~
143
4
4
12 2
+1Oele2e 3 e4 ell + 12ele 3 e 5e612 + 6e5le2e17 + 6e4e8 - 11ee 2 e4 e6 eg
-2e5e 4 e615+ 7e83+ e2e 14 + 8e3e 2 e 4 e 5e0 + 5e4e23e1 4 + 2e2e3e 14
+8ele3e 1 4 + e6le4 e1 4 - 12e1e3 e6 e 14 + e4e2 e4 e6 8 - 1e2e4e 12
2e2e 6 e8 - e2-4e 6 e 8 - 13e 2 e3 e4e7e8
+25e e2 e9e4elo
2 4
2
2 2
3 2
1 2
1 3 2
ee2e 17 + 9e2eselo + 6ele2e4 el0
+18e 3 e5 e 6elo - 20ele 2 e4e8 e9 -3e2e2e 6 e1 2 - 12ele 3 e 6 e12 - ele 23 + 8e1 e4 e8 ell - 7e4 e5elo0
e2
4 e e - 5e2e56e9-7e
9 -eee4eee e6 e 9 - el2
ele 2 ee6 e9
4 5
+3elel23 e 4 e5 e8 - 8el4e
5 6 9
3 e17 + 4ele2el 7 + 3e3e6 e9
+7e2 e e8 + 4ee 4 e 8 + 8e2e4 e8 4e4le
- +ce
23elee
5 el12+ 22e l e2 e5 e8
-2e e3e e + e2e20 + 6e3e265 e12 2
22
C2
+e2
4
+ 5e2e2e 6e8
eeeee-2elee6ee8
-4e3e 4 e+
e
-9
~~~~~~22
+7e2 e2e9ell + 6ele2eae5e8 + 4el2e2ee6e9
2
+C4~~~2
7e8
e2e2
2 2
+ee3e5e6 + 9ele2e5ee7+ 15ele2e4e5e9 + 3ele4e7
-15ele 4 e5 e 6 e8 ele 2e4 e6 e8 -elle
3 - eleee 17
+ 3ele2 e 7 e14
+ele 2 e 3 el2 + 7ele3e 5 e-11 11le2 3e5eil + 7e4le4e5ell
+4e2e 4 e5 ell - 2ee 3 el7 - 8ee 5e8 e9 + 7e2 e5e8 e9 + 14ele6e8 e9
e4 ee8
1eaeses
9 - 5e4e 3ee9
8 e + 3e5e2e
+13e4e2 e 12
12 9 + 14e33+
9-12ee2e2ele
+4e2e 3 e4e9 + e3 e4 e1 7 - 4ele 4 e5 e 7 + 3ee 6 1 0 - ele 7 + 4el3e4 el7
e + 4ele3e8
-e2e
2 e3 e5 e12 - 7ee2e 3 el10 + ele6 17
6 -16ee
3e5e6 e 13
e13 - 2e6ee13+ 7e2e
-5ele3e2e9- 22ee3e9ell
4
5l2
3 5
4
ele 4 e6 - 7e 2 e4e6 - 5ele 5 e6 4- 2e 2e4e7 + 4le2e 4 el2
+ 6e2e 6 e8 + 2e4le2e8 + 13e2ee 8 + e4 ee
- 6eee 3 e6 e8 - el e2e3 e 5 e 9 + 9e2e2e 1 2
-7e2e3e4e2
+8e 1 e 2 e 3 e
8
5ele2e 7e 8 + 9ele 4 e 5e13 - 6e2 e4 e5 e13 + 7e3e2 e 6el3
2ele
_e~~~e_
2 e2
-6ele2e 6 e13 + 8e2e3 e 6e13 + 3e2 e 3 e6 e1 3 -2e 2 5e12 -2e523e, 13
~223-6e22
-3ele 4C5e5 eio + e2e4e5 e7 + 7el3e2e 5e1 4 - 6elee 5 el4 + 8eje 3 e5e l 4
-3e 2 e3 e 5 e14- 4ele 4 e5 e14- 7ee 3 e 4e 14 + 7e2e2e8 - 7ele3e 8
~~~~2
2 e + 6ele e4 e6 ell
e2 e e es8+ 9e e2e3e
2
4 9
l
-5e4e 2 e1-58 -e4 17ele
2 5 7
+3ele 2 e4e
-
el
+333234e4eae7
-5C~~~~~~~~
5 23
2
2 3 e~9,
-4e2e4e 8 + 6ele2el18 + 18ele 2 e3 e4e14 + ele6e 7 + ele2e 4 e5 e8
-5e3 e4 e5 e6 + 3e3e3e18 + 14ele2e7el0 - 13eee 7e10 + 4ee 3e7e 0
+ 6ele3 e4e8 + 36ee 2 e3e5 eio
-15le, 2 e3 e4e1o - 2eee
-ele42e5 e9 + ee 3e2e9 - 3e2 e3 ee 9 + lele 4eCeg
+e4e 2 e5 e6e7 - 5ele2eloell + 8e2e3 e4 e5 eIO + 5el3e2e4e9
82
36661 7617
42
-eC1 2 e3 Ce
e1 +±3e2 e4 e 8 2 + 11e1e 2e5e1 6
5 e- - 3ee 4 e 8
-9ele 4 e 9 e 0 + 5e2e 2 e20 - 2ele 3 e20 + 18e2 e 3 e 7e 10o + 12ele2e 7 e1 0
I'l-132351
36264618
2 3
222
3 e2 e6e7
6ele2 6e7
-13e 1 e 3e 4e 5 e 11 + ee3e
13 -3el
e6e
9ee
e
e6e
11ee
+4e 3e66e7+ 9e2 3 7
4
7
2e3 e4 eI 2
+el8- 2e 1e2e5e 6 - 4ee 2e5 e7 + 6e e3 e 6 e 9 + 6e4e73e6 e9
21661
261236
3
2 666e632
66
+19ele3e 4 1 2 - ele2e 3e6e10 - 2ee 4 e5 e6- 8ee 7 ell
- 6e3 e5 e7e 9 + 4ele 2e19 ± 9e6e2 e 3e17
-6e6e 4e7 e 9 + 1 1e3 5 e 7 9eg
4ele3 e4 e 7- 8el4e2 e8 e 10 - 3e1 2e19
-6ee 3 e 4 e6 e 0 - lel-
3~se
,112e2 636Sl 41
+14e12ee 8e610+ 11e1e 3e 8 elO + 5ele 3 e 4e 5 e 8 + 51le 2 e3e 9
2 2
e
-9e2e 5e7 el0 + 4e 1e6 e 7 e1 0 - 16ele2e
3 e5 e1 0 - 4e 2 e3 e 5e7
-2e6e 2e46 + e2 e3 e1 9 -22e 2ee 6 el10 + 16e3e 2 e 3 e16 -15ele2e
5e2e3ell I+ e3e2e2e
+5eme3e4 e26 e7 + 2e 4e
43
1 3ell +5263
6162657
ee2 e8 e
3
4ee2e 6 e 0 -14ee 3e6e
e24 -3ee
3e 16
6e 2
+2ee 6 e-6ee
0
2 e4 e2e 7 - 3e4e2e7 e9 + 5e4e e4 e-1
ell
-e 6e12
e
e
e
5e
5e3ee
{7e,262eI6 -7e 2e3 13 - 7e
4
1 2 3 51 3
2 3 32
e-e1-5eee
-7ee
3
3 e9 e 0 -
6ee
e4 e9
2 e6e1 0
-18e3e
3 e4 e7 e8 -
+ 12e4e2 ee 1 0 ±1e-e
3e6e 0
7e2e 2 e3 e15 ± 7e~e3 e5 e1 2
e7 - 27ele 2e3e7elo
-6ef e2e52e 2 - 14e3e4e_ 5e- 2 e2e22 + 7e2 e4 e52e6516341
e
e
9ee
e
e
9e3e2e3ell
-6e,1 ell 4- 3e5e
3
4
13
2
136 1645
6667
1,
66¢
27 2
+4e]ee
6 e7
+ 8e 3 e2e 6 e 7 + e 4e 2 0 + e 2 e 9 e 13 + e 2e 2e
, 1,6[713
-5e"e 6 e
1 ,
0
-2 -416 67
3ec2e2 e e7 + 7e3e 2 e7 el2 + ee2 e7 el 2 + e 2e3 e 7el2
15e2
6e 4 e25e
+8ele3 ee 5 e7 - 3ee
3 e5e
- 8 - 5e2 e4 e2
1 e8
1626369e10
2
8 8 +
662el6o
-612622_2
e
ee
e
e
ee
-8I'eele2eae
e
9e2e
e
e
e
11
6as1
2631
31 1
- 3 4 6 7 762 626478-- 7:!3604
+17ece 2 e3el - 2e4 e6 e7 + 42e4e6- e4e2 - 11ele2e
3e
+17et 2 e3 e 4 e5 e 0 - 9ele 2 e5e6 e 0 + 19ee3 e4 e 5 eg - 21cle2e 3 e7 e9
+6ele2e 9 + e6 e 7 ell - 3eMe 8 ell + 13e3e2 e5 e6 e8 + l0ele 3 e8 e 12
-2e 1e 2 e9 e1 2 - 13ee 2 e48 el2- 7e
2 3 e8 eI 3 +233e2 e2e 4 e12 + le2e 2 ee 1 2
-
+2e3 _e4 e5 e 2 + 12e1e3 e 6 ell -7e2e 3 e6 ell + 16ee3e 6 eI - 7ele4 e6 el
-5e3 ,e4 e6ell + 1Oe2e5 e6 ell + 4e2 e5 e6 ell - ele6ell + 9ele3e3 e4 el0
e ell - 9e2ee 5 ell - 6e5e
-1e2e2 2 e4 e 6 + e5 e7 e12 + 5e4e2
3 e5 el
2 1 2 5
2
+ele 2 e4 e5e6 - 7e7e8e9 - 3ee9 + 2ee 7 e9 + 12ee 2e3e4e7
7e 2 ee 5e 8 + e e2e4e2 + 5e 3ee 9 - e4e7 e9 + 13ele2e2e 9
2 2
3e3e
2e~e6 e7 ± e e e - 2e2e4 e2e 6 - 7eje2 ell + 2e 6e 7 e1 l
5e3125
+3e 3e7 ell - 2~~~~~
3 e 4e 6 e8 - 2eje 2 e3 e 2e 8 - 24ele 2 e3 e8 e 10
+8e2e 4 e6 eI2 + 2e 2 e4 e 6 el2 - 14ele 5e6 el 2 + e4Lee14 - 3ee4e 14
+2ele 0 e 3 -7e3e
+ ele2e 3 e4 e5 e7
2 e3el 0 -21ele2e3elo
8e2e2 e4 e7 e9
3 8 -37e 5 ee 7 -8eee
4ee2 ee 8 -8ee 3e2e
5 e6 e8
+3e 4-
+4e~
22238
8eeeC
C
2 e7-8l2e5e8
83
8
2ee7
5 3e2 -ee 466
e~3 2 2e1e
3e4e6e7
e14 + 6e2e8
+ 8e3e7~62
-2e2e8 e~~e
l1e3 4 e7 e 10
6ee6
2e5e13 2
e e7 - 8ee + 6e2ee8
23 -3ee5e7eio-- 2ee2e
ee6C7-4 4 6
_362656761O_
3e
+
1e2e3 e7
13 8 e12 - 2e1 e2ee 15
-6l3~,3+
+6e2Ce2e
-616265613
3 5 e13 + 4e41e
5 e 3-6e6e
e 2 e1012 - 2e elOe12
+
e 5 e9 - 4e32e
+3ee3
4 e5 e9
e
e
~
2
23
8 e 12 - 6e3e4e 9635
-5e 1 e 2 e - 3e3e
--52 e3e 5 e 8 + 2ee 9 e12 -3e 4 e3612
+3e 3 e6 e5 - 3ee 2 e5 e1 2 - 7el1e2 e 4e 17 + 5e1e2e3e 4e9 + e6e8 e1 0
5e4e 2e 12 -4e 1e 2e 5 e 7e 9 + 8e2e2e 7 e9
+ 4
+12ele 3 e4e ±
4 16
+ 3ee 2 e4 e15 + e2e5e6
+2e5e 3 e7 e9 - 2ee3e 7 e9 + 14e2e63e7e69
+4ele2 e 2e 6 e
ee 4 e7 el 0 C3-
e e4 e
-13e
4,9-- 61111
-[-612636,5e6
e32
3651
8
e2
e2
+26 2_
-17ee5e02
e3e5eo
18e
ee6 - +9ee
+4e
2e3ee918el3ee4e
e4e11
3e2e
3e1 ee -
1~~
+6e3 e 4e e 7
-9ee 2 e4e
-3e06 2 e 3 e6 e
4
- e5e19
3
7
6 7 - 8e 5e6 7 + Oeee 6e7
8
e27e
4e6 e 10 + e24ee
- 5e2e 3e 4e6 e 9
1 0 - 12e26e
4
22
4ee5
e 2 e3 e6 11
ee,9+e2
8e
+14e12e3eI0 + 616ele
5 el68- 3ee2ee 6 e7-18eee
23e
21e
e
8 e 9 + 6e6e 4e 8e9
1
11e4e
+1Oele2e 8 e93 e8 e9- 13e2e 3 e 8 9
-~236e461
2
2e
3 2
~
2ee1 0 - e2e2 0 + 32e 3 e4 e13
-2e4e C4
5e 11 - ele2e 4 e5 e8 + 6e 5 e9 el0 Ce 4C
-2c2
2
C~~~~~2
-3c,
-3e1 23e4 e13 + 5e1 e2 ee 1 3 - 2e2e 3e 19 - 8e2e2e 4e7
+5e2e2e 3 e 7e8 + 2e2 e5 e17 ± ee5 e59 + 6ee22e4e15 + 3e3e2 e3 e7e 9
+2e3e2e7e8 + 9e 1e2 e4e5e8 + 43e 4 e6 e 8 + 8e2ee 6 e8
- 72e3e4e15 + ele2e 3e14
-3e5e 5 e 6e8 - 7e2e3e4e9- 4e 2e3e4e9
22
-2 2 _4
e1 e2 64 e14 + 3e6e 2 e4e0
-1961e 2 3ee14-e16-2e~2ee5,
eel2
2
+7C~~~2
-2ee13e 4 e5 e7
+7e22e
3 e4 e5 e8 + 6ele 2 e 3 e5e 7 - _
3
25
+e2
+ee 2 e 4 e5 ell - 3e5e 2 e 3 e 14 + 8e1e2e 3 e 14 + 1Oe1e4e 5 e10o
-4ele2e 4e6e7 + 1961e2 63 e6 e12 -4, e 1e46e7e 8 + 6ee 5 e7 e8
-9e2e 5e7e8 + 8e4e 5 e7 e8 - 4ee 6 e7 e8 + 4e3 e6 e7 e 8 + 21eje2e3 e 5 e11
2 e413- 4e2e2e40 - 5e2 e3 e 4el5
+ele52e4e9- 3e32e
8eo -e3e
e2e 4
_ 2
+7e8 e16 -4e2e
2e 3 5e6 + 3e3e 5e8 + 15e1e2e3e4e6e8 + 7e3e 10
e9 + 7e2e8e10
- 3e6e
+ 4ele 5 e7e
16 - 5e3 16252
+7e2 e4e1 o0
4351
73 I
3~~~
-6e62e2 e5 e15 - 5ee2e 3e12 + 2ele1e13 - 5e3e4e613- 73e 2 e5e9
_4ee2e2e 9 - 3e2e4 e14 + e1 e4 e14-3e5e 5 e14 - 7e6 e8 e10
2 2 31636e9
2
-3
e2e3
C4 2
313e6 e9
2
+4ee2e
3e13 - 7612e3 e13 -- 3e
-4le42e 6 e9 - 3ee 2 e4 e 13 + 8e0e e4 e1 3 - 7eje~e4 e 13
22
+8e4e 3e 4e 13 + 6ee6e 9 - 5ele 3 ee4 10 9e2e 2 e5 e6 e 9
62271
__+7e.e2 e3 e4 e6e75e2e 2e 3e 5 6 e8 6e4e 2 e7 e-1 1 e2e 2 e7e
+e 3 e37 e11 - 3e2e47e1ll - 6e 2 e4 e7ell - 16ee 3ee612 - 8e3e4e
± ee 6 el2
612
2 3
2 9 ± 3e1 e3 e5 e15 ± 4e12ee3e12
+2e6e3 e1~~~
5 e6e7 ± e29
+ 4ele4 e 3e 12 + 4e 1 e4e4 e 1 1 + 15ee 2 e 4 e 7 e610
+2e2e25e26
2
~~~~~~~~4
2 5
-e 2 e3e 4e 5e7 + 5e5e9e10 ± 13ele 2 e 3e 7e1 1 + 7e2e3e1 0
-14ele 2e4e 6 e7 + 4e 2 e 3e4 e6 e9 - 6e3e6e 9 - 3e3e 2e3 e8
2 1 8 264567
4622
e2e2e2
4 2 2
e4 e5e6 e7
+e4e13e7 + 2e1e3e4e8 + 4e 2 e3e4e8-12e
7e 2e 4 e6 e9
+
e
le
6
e4e
e
e4e
8e
1 5
9
-9ece3 e9 +
2
- 2ee+
9 e2
e1346e7e9 1e3e2
-e
2112451
2 +e451
-- 61e6364e769-- 6126367
84
ee
5e7
0 -6ee
-635
7 1o-- 64~,
-4e'le 2 e3 e6 e9 - 6e2e4 e8 el0 + 6e 2 e4 e8 e10 + 7ele 5 e8 el 0 - 3ee2 Ce20
-e4e. + 3ee 2e3e5 e6e7 - 1oee 2 ee 8 + eC3ee 8 + 4e.e2e2elO
C2
2
e2
2
-16ele 2 2e4 ell +
45ele
2 e3 ee 9 - 4ele2e 3e4e 8 - 4ele 3 e4e5e 6
-7ele 2 eell
-
3e5e2e6 ell + 6e3ee 6ell + 4elee 6e11
-155ee
e2
2 e 4 e8 ± 4e1e
2 8 2
-1
82
e
85
Table of Pm,4 for m > 3
m
Pm,4
4
2e4 e6 ± e1 e3 e6 + 3e l e4e7 + ee
2
7 - e2 e + e4 e5 e7 + e2 e5 e77
-e 2 eCe8 + e 6 e1 0 + ele 7 e 8 + 3e 2 e 6 e 8 - ee 6 e 8 + 2e2e 4 8 - 2e 3 e6 e7
e3e1 3
- 2e 6 e 9
9 leee
+e2e3 e9 + 3e2e 5 e9 + 4e 3 e4 e9 - e
-2e 2 e 5 e9 + e3 el 3 - ee10 - 2ee 5e 10 - 3e2 e3 e4 e7 - 2e2 e3 e11
-e2e 3 ell + ee2ell + 2ele 3 e12 + ele4e11 + e2e4e10 -2e 1e2 e 3
-2e 4 e1 2 - el e3 e5 e7 + ele2 e6 e7 - e1 6 - e 2 e1 2 + ele15 - 3el1e 2e4e9
23
_ 2e
4e4e
o- 8 + ele5 + e5e11
-3
22
-
+ele 3 e4e8 - e,2 14 + 2ele 2e 3elo0 e77+e9
-2e1 e 2 e 5 e8 -e 2e6
-3e2le6el2 - 2e3 e5 e12 - ele5 e2 + 3ele 7 e1 2 + e2 e6 e1 2 -
22
5
e 12
e2+
e2 e3 e5 e 6 -3e
e8 + e 3 e 5 e,8
e2 e1 4 1e4 e 5 e6
C2
5e 6
le17
+2ele 6 e1 3 - 3e 2e 5 e1 3 + 2e2le5 e1 3 + 3e 3e 4e 13 - 3ee3el 3 + 2e2e 3 e1l3
+ 2e1 ~e 1 5 - ele 2 e 15
-e1le2el3 - 2ele4e15 - 3e 2e 3 el5 22 +3el5ie
2
32
-ele 3 e4 -4e~e 8 ± elee4 ± 4ele2 e5 e2 -3e~le 2 e16 ± e6e27
-4e1e2e3 e2 + e4el6 - ee 19 + 4e2 ee 1 0- 2e4 e6 e1 0 - 3e3 e7 e 0
+21262e31l3 + 3ele 2 e27 - 3ele2e4l3 + e2 e1 6 + 2ele 2 e17
5ele 2 e6 ell - 3e2e7 e9
2e+3e2e 3 e5 e8 - 4ele6e5 e8
5 e1 1
+-e5- 2e4e12 + 3ele2e4ell - 4e2e3 e 4 e1 1 - 3e 2 e3 e4 e1 1 - 3e2e2 e1
+3ele 3 e5 ell - e 2e 18 e1 5e8 -4e 2 e5e8 + 5e 3 e4 e 5e 8 - e 1e 2 e3 e1 4
e
+5ele 2 63 e 6 e8 + e1 2e
3 e4 e9 - 5ele 2 ee9 - 2e2e4 e 5 e 7 - 2e 5e 7 8
eel2
e
+
2
+e 4e2 + 2ele 2 e4 e6e7 - e9ell + 3ele 8 e11 - 2ee 42 1
4 e 0 - 4e4ee 6 + 2e2e2e8 + 6eelo0 + ee 4e 0
- 2ele 2 e 5e6 - 3e 5 e6 e 9 + 3e 4 e 7e 9
-2e 4 e 16 - ele 3 e 4 e5e7 + e362614
-eale 8e 9 + ee 9 + e 2 e9 - e32e5e 9 + 4e3e5e 9 - 3e1 e5e 9
-ele3elo - 2e
+2
l22
+2e1 e 2 e 5el0 + 3ele 3 e5 e0 - 2e8 e12 + 5e 2e 5 e 6 e7 - e 3e4 e 9
-7e 2 e3e 5 elo - 8ele 4 e5eI0o- 4e26e2e6e1o+ 4ele 3 e6 e10 + 3ele2e7ei0
-
2
+
562636768 - 3ele 4 e 7 e 8 + e21le 2 e 8
e2le3 e7 e8
+2 e e e + el e e e
3eW1 e3el
C2
2
12 e 1 2 4 1 2
1 3 4 12
+4e 4 e 5 e 7 - 3e 5e - 3 23
3e2le
+
e
4 e7
2e2e 6 e 8 - 2e2e 3 e6 e 7 + 3eee 6 7
-e12e+ ele 2 e3 e4el0
-4e 2 e4e7
2e12- e3 e17 - 3 3e 4 e6 e7 - 46le 5e6 e7
e2
-2e 1 e 2 e 7 e 8
24
2
+2
C2
+3e 2e 6 eo - 2ee6e7 + e1 e5 e7 + 3e 2e 4e14 + e2 e3 e5 e7 + e3 e8
-2eje 3 ei- e3elei - 4ee 4e6e8 + e2 e4 e6e 8 + 5ele 5e6 e8 - 4ee 6 e 8
22
23
2
e2
-4e 3e 5e 7 - 4e2 e3e4 e8 + 4ele 3 e4e8 - 2 e3 e1 1 + e2 e3e 0
+ 2
2
+ele3 e5e6 2+ e2e4e6 + 21le 3e16 - e7 el13 ± e8 e1 0 - 2e2e8 e10
ee
2e2+ 3e3 e5e2632a2
e1 4e14 - 2el1e2e3e5e9
-e6e
-3ee9e0 +ee
- 3e2e 3e 6 e9 - 3e 2e 3 e 6 e4511
9
+5e26e4e 5 e9 + 2e 2 e4 e 5 e9 + 3ele2e226
6 e 9 ~~~~~~2
616
e
2e
e
2e
2e4e6
+
+
2ee
4 5 e2
+2ele 4 e6 e9 - e3e14 + 4e5e15
2e2e7 e1
2e3 6 e11 -+
7e11
+4eee1 1 -2e51 + 4e 6 e11 42e
e
e
ee
+
e
2 7 9
+2e 0 + ee 1 8 + e20 - 3el1 e2e4 e 5 e8 - 2ee 3 e4
e
e
32e
+
2ele2ee1
+
e3e9
e
9
4
3
+ele6367e9
2
86
6 e4e2 - e2e20 + 4ele 5 e 8 + 3ee 1 1 e 1 2 - 2e5e 9el10 + 3e2 e10 el 2 + 3e 2 e 4 e18
-4ee 2 e 19 + e3 e21 - 2ee 2 e2 1 - 2e1 e3e 20 + 3e e2 2 0 - 3e1e4 e19
-2e
2 e 3 e 19
+ 3e2e
3
e
19
+ 3ele2el9 + e4e 3 el3 -
5e
e 3 e4e
1
7e 9
+ 4ele 2 e3 e2 - 3ele3e 6 e11
-3ele 3e4e2 - 2e3e6 e9 + 8e3 e62eg9
+2e: 2 e3e13 + 2e2e4e5e61 + e3el 1 3 + 3e2e 7e1 3 -e 2e3e 0
+e24e 7 e 9
-
3e 1 e 2 e 3 e5e8 + 3e2e5e
22
2e2 e 3 e9 e1 0 + 3e2e2e 5 e13
8 -
-3ee 3e 53e 3 + 8ele2e8e1 3 - e3e 5e8
-61
_
e7e 0
-
2
3
e23e11
-e, 1e 2e 3 e 4 e1 4 + e2 e 2 2 + 3ele 4 e9 e1 0 + 2e 8 e1 6 - 7e162e 3ee1 0
e eg + 2e2e2e
-3e' 7e 8 e 9 - el e4ell +3 lOe2e
5 e 9 + 4e 2 ee4- 5 e 9
~~216469
3 4 5
2e2e-2c,
-2e2e 9 e1 3 - ele 2 5e 9 - 3e2e 4 e5 e9 + 6ele 2 e3e6 e9 + 7ele 2 e 7 e14
e e + 3e e
+2ee 2 e5 e2
1 4 - 6ee2e 5 el 4 + 3e e 3e 7e8
32
-
-2ee 1 6 - 14ee 2 e3 e6 e12 - 3e3e2e,13 + ee3e 8 + 5e3e2e13
e2 e 0 + 3e1 2ee8 e9
+7e:,e 2 e8 1 2 9e1 e3e 8e1 2 -7ee 2e9 1 2
+4e 5 e6 e 13 + ee1 1 13 + 5ee
-2ei[e 3e 9 e 11 + ele2e,10eii
-e 2 e 1 4 + 3ele 3e20 + 4e l e
6ee
46111269
2 e e2
e2 - 2e6e
2e9ell
4~~~~~4
1182 4
5 ell + e42e
- 8e2e 3 e 5e + e 5 e1 9 - e,2e2 2 + ele3e 4 e 1 0
7e 9
e22
e9 - 2e2e4e92+ e2 e4 e -7ere 3 e6 e1 3 - 4ee 4 e6e 13 + 4e 2e2e 7 e1 1 - 4e e2 e 6 e2
-2e]e2e 3 e6 eo0 + 5e 5 ee 7 - 5ele2e 5 e 6 e 8 - 9el 2 + 4ee 3e6e 7
-5e 2,e3 e26e7 + llele 5e6 el12 - 5ele 2 e7 e1 2 + 7e 1e2e 7e 12 + 8ee 3 e7 e1 2
2
2
e2
2
-7ee 3 e7 1 2 - 4el1 e 4e 7e 1 2 + e3 e5 el3 - 2e2 ee + 4e2 e4e 8
+ 2eje2e
3-- 5 e8
2- 4 e 15 -462e 3 e 4 e1 5 -3e 2 e 3 e 4 e 15 + e 2 e3e
21
-- ee,-e23e
8
+ele 2 3 + 5ele 2 e5e1 1 + 2ee 3e 8 e1 1 + 8e 2e 3 e8 e1 1 +- 2e l e 4e 8 e11
-3e 1e2e
13
+ e3e4 e11 - e3 5e7e 9 + 4ee 6 e7e9 + 4e6 e 7 e11
2ele3e4 e6 e7 + e2e4 e7
+2ee3e 7 e9 + 6ele3 e4e 5e1 1 -
2e 22ee
22 2 - 3 e4e7
- 2e1 e 2 e3 e5 e,13 - 4e6 4 e6 e8
+3ee4e7
2
-2ei'e
4 e52e8
5 6 e1e1
-
-
2
e2
2e3e 4 e7
32,e42e
6 e8
4ee2e 3e 7 e9 + ele2e 3e 5e1 1 - 2e46e
±-4ee2 e5 e 7 e8 + 2e12e1e5 e1
-5e"e
2
3
2
.22
-
8e2 ee
e
511
- 4e3e4 e 6 e11
- 7e 2 e 5 e6 e1 1 + 4ele 2 e 3 e 4 e 6 e 8 - e3e 4 2e7
+ele23 e5 e27+ 5e l e26e
1 1 + e3 e 7 e 8 + e3 e53e6 + e24e16 - e2e 2 e2
+ele 3 e9 - 3e2e3 e6 e9 + 11ele2 e6 e 7e 8 - 362e4 e 7 e 9 - 23e7e
2 24
e2 e25 e20 + 9e 4 e22el - 3e e e 12 + 6ee 7 e13
+3ee4-36¥62646769~
2 e4 e7 e9
-22e2 ee
12
+ e1e4 el4 + 2e2e4e14 - e2 e2el4 + 2e,36el5 + 8ee 2 e6el3
+6ee 3 e6 e 3 - 3e 3 e 8el3 -e2e 2 e 3 e 5 el2 - 5e3e3 e7 e1 1 + 2e2e 4 e 7 el
-3le2e 3 e 4el12 + 2e 1 e 2 e2e 4 e11 l- 3e3e6 e 1 2 + 7e~e 6 e1 2 - 7e 2e6 e 7e 9
- 3e 4e7 e1 3 + 4e1 e2e3 e1 4 - 3ee 2 e3e 1 4 +- 7e2e3 e5 e1 2
+ 7e 2e 2e3 e 7e1 0
+2ele2e
4 e5 el2 - 12e3 e4 e5 el 2 - ee
5e12 + e63e
e2e
+
5e
el
+4e 2 e 4 e 5 e6 e 7 - 2e2e2 e 5 e 6 e 9 - 2ee 10 1 3
3 e2
l
-5e 1 e2e 4e7 e 8 - 22e3e 4e1 2 +-2e 2 e3e 4 e12 + 2ele 2e4e l2
+5e 1e 3 e6e1 2 + 4e2e6e
1 4 - 2ele3e 5 e1 2
5 e9 - 5ele3e5 e9 + 3e3e 4 el
e2
e~2
2 2
2 2 2
e5e6 -e 2 e5e6 - 4ee17e8 + 4e 2 e7e 8 + e2,e3 e 4 e 5 e 0
-2e_2-14e2 e 2 e 7 e 13 + 7e 1 e 3 e 7 e1 3 - 7e 2 e 3 e 4 e6 e 9 + 2e2e 18 - e4e20
-6e3e
4 e 3 -ee3
2 e4 e6 e1 -2e~e~el-9ele2e
7
- 29 ± 7e1e4e 5e2
3ee4el3
- 2321 -66162e4611
-5e3e
8 e 13
87
e e14
8 + 4e 4 e6e 8 - e e 3 5
eo - 5e3e 2 e6 e1 3 - 3e2ee
+2ee 2 e6e 5+2ele2
4612
3 2
2 6
3 1
4e 3 e1ell
eee+
+7e 2 e 3 e 5e14 - 3ele 4e 5 e 14 - 5e2 e3e4e 9
263362
~~~~~~~~
33~~~
+ ele e4e5e2 + 2ele e e 7 -e3e4el
+2ele2 e3el2 - ele 5 e7e9
2 4 1
6
0
2
+3e24elo + 6e2 e4elo0- 4e 2 e4 ee8 + 4e12e6eel2 + 2ee 3e 6el2
2
-ele 2 e4 elC3 - 9ee 4 e6e12 + 15e 2e 4 e6e12- 12ele4e 5e 1o
2
2
-2ele3 510 + 7ele2e6e14 - 6ele13e6614 + e1e2el5
-26l1e25610
-2e312
3 2
2
~~3
4
-e 3 e1 2 - 2ele 9e 14 + e 1e6e 9 + ele2e 1 7 +3ee 3 el7 + e3e4e17 -3e 3e 4 e17 -3e2e 5 el7-
1-e4ee7
1 7 - e4e13 e 7
4e e 2 e 4 e6 elO0
3e2e3e4e5es - 6eie 2 e3 e1s + 4e2e6 e7 -2e 4 e6 e7
2e2e3e15
6ele5e6e7 - 23631
_2 2 6e8 -__612667
24 -e2ee
63 -2e6
_] e
-5ele_5123
2 e7 +
1eloel4 -
e7
-e2 e3 eei1 ± 4e32e
8 e10 + 6ele2e3e5e6 e7 + 2e4e20 + 4elee 5
+ 3e 5 e 7 e1 2 - e4e 8 e6
1 2 - 3e2e 8 e 12 + 4e 4 e 8 el2
+2ee9e12-3ee13e16 + 2e2e4e2 -2ee 5 el3-ell
e15
+8e2 6 3e 4e7e8 - ele2e4e5e8 - 2e3e15 + 8ele 4 e5 e6 e8 - 3e2e2e3
+6ele 2 e3e15 + el3e2e 4 el5 - 2ee3e 6 e10 + 6e 1 e 3e 5e 7e 8
2
e3e7 - e24
+ 5e 134e9 - 5ele2e6e + 3ele7 +2363
+562e3649
+2e 6 e
1
1 466 9
±e22 3e 4 e6 e7 - 3e3 e 4
-2ee1 62 e 3e4e
e2
7
5
6 + ee 9 -
e6 e14 -e
2e20
032
- 4e 3e6 e768 - 3ele 2e 3e6 + 5ele 2e4 e 5e8 + e9 e15
1
+ 6e3e4 e6 ell + 4ee2
-4ele 2e 5e + 8e2e3e6e -ee 3 e6e1
2 5221
2
3 2
2
2 ' 2 e6e8 + ele3e15
+ 5e1e2e4ei
3ele2e3e13
+4e 2e63e7 e9 + 4ele 2
+8e2e5e6e9 + 11e2e3e5e6 e8 -Oeee 7 e8 - 5le 4e6e7
-ele 2 e 3 e8 e10 + ele 2 e3 e6ell - 3e1e2e7e10 + e2e3 e7 e10
3
-7ele3e7e10 - 3e3 e4e7el10+ e le21 -3e2e4e5el3 + 86le4e5e63
e e14
-4e2e 6 e14 + 3e4 e6e14 - 5el3e7e14- 2e3e 7el14+ 4e1e 8 e1 4 - 3e2 8
e7e8
-5e 3 ee9 + 4ele 2 e 4e 5 e2 - 8e2e5 e 7 e8 - 3 4 e 5e 7 e8 - 5e3e6
+e 3 e4e5e8 - 3e 6e1 6 -8el
81 e 3e6e26
9e4e5 e6 e 9 + 2e2e 4 e7 e11
-4ele 5e7elI + 2el3e2e8 ell - 8eee86e11- 9e 2 e4e5el
e9
-3ele 2e 3 e4e5e9 + ele 4 e8 e9 - 4e3 e4 e 8 e9 - 3el2e5e 8 e9 + 8e2 e5 e8
e769
e
e
e
1 3 e4 e14 ±+
12
2 5
± e-6e
-4ele 6 e8 e9-2ele2e4e14
-5ele 2 e6eg + ele9 - 3e2e5 e6 e7 - 3e3e5 e6 e7 - 3e2e3e5e0
2 e4e6
-4ee 3 e5 e6 e9 + 6ele 3e5e 9 - 6e 2e 3e5e 9 - 3e1 e4 e9 ± e
e
+3ele 5 e + 2e l e3e1 4- 2e1 e 2e 4e2e7- 3e 3 e 4e11 - 3e 5 e8 11
-e41le9ell + 3e2e9ell + eel0oell - 2e3 e1 0 e11 - e2e2ll + 5e3 e4e6 e7
ee 5 e6 e10
+e e9 + 8e 1 2e4 e5e10 ee 5 e0 - 4ee 4 e6 e0
2
2
5366660-2e2
-10 +462e1
4e2 e e0
~
662666 --2e2lei
- 6e2e4el2 - 6ee6e10
-5e 3 e5 e6elo
+2e2e 3e 4 e3 + 16ele3 e 4e6 eo0 + 3e2e9 el13 - ele 5e 15 + 3e2e5 e15
e + 2e2e 6 e10
+2e 4 e 5e 1 5 + 4e3 e6 e5 + 4e2e 7 e1 5 - 2e2 e 7e615 - 3ee2e 4 1 3
+2e2e3e 7 + 3e 3 e 4 e 7 e1 0 + 3e2e 5 e 7 e1 O + e20e5 e7e1 - 5eje 6 e 7e1 0
C2 + e3D
e2 e2
3 2
e8 e 03 e-8 ee9222e3 e7
- 6 5ee
e83e
e4e6e9
9e1e
- 4e 2e- 6 e eel2
-2e
2 ele 5+
+2ee3e7
2 e4 e6 e + - 4ele
63e8
+5ele3e 8 e9 + ele3 e5el5 - 7eje2 e6 el5 - 2 e 4e 5 e6e7 - ele2e16
88
-3ele 8 el5 - el e2 ee 6e7
+ 4ele3 e6 e2 ± e3e8e 0 + 8ee 2e 3e4el3
-5e'e2 e3 e8 e9 + ee 2e9 e 0 + ele2e9 el0 - 4ee 3e9e 0 + 6e3e9e 2
e2
-2e:eloel2- eel 4 + ~~~~2
3ele2e5e6elo
0 - 3e2e3e1 4- 4e 2e3 e4e5 el0
-2le2e+ 8e1
- eie0 3e8eIO + e
4 e8 e1 0 -9e 2 e4 e8 e10
+ 2ele5 e8e 10
+e2e3 7 - 5e6e18 - eel 8 + e3e 6e8 + 3ele2e4e7e10 - 3e4e9 ell
+2e:3e 2 e3 el 6 - 3e2e 2 e4 e 16 + 8ele 3 e4 e16 + 7l2e 3 e 4 e7 e8
-ele 2 e3 e7e11 + 6ele2e 4 e 6 e9 - 2e2e4 el 6 + 2el3e5 e 16 - 3e3 e 5 e 16
+5e 2e 6e1 6 -3ele 7 el 6 -2e 2 e5 e17 + 4ele 6 e1 7 -el4e 2 el 8 + 3e e2el
2ee3 e -2e +2cje~e,18
e4 e 8 -2ee6e8
- 2elC4Cl8 3elCelS
-
89
8
Table of Pm,5 for m > 3
m
4
Pm,5
-ee26e7+ 4e5e6e9 + 3e6 e1 0+ 4e1 e4e5e1 0 -2e 1e3e6e10 - 73
+2ee 5e9 - 4eee
+3eel2
-e2e
e4e8 -- le19- e2 e8 e 10 - e6e8 + e3 e4 e6 e7
9
ele 2 e2 el2 + e6 e
2e 3 e6ell - 3e 5e 7e8
7e612+
-2ele
e2 4 e14- 2e2e4e14 + ele5el4- elee15 + e1 e3 e5
2e2 e3e5
eee
6 3e1 +2e1e2e17 + e2e4e5e9 - 3e2 5 6 7
-2ele4e5 +ee 2e6
1 - 2ee
+ e2e6 el2- 2e2 e6 e12 - e6e e ± e2e4 e10
1 1 - 3e 3e 4e 5e8
2e2 e3 e4el + e1 e3 e5 e1 1 -2ee 62 e e1
-e 2 e6e2~~~~
1 0 ± 3e8 e12
+ 62e6e8
-- 6129 + 21e6261463 -2ele263e64 + 362e26e8
+e2 eel2
4 el2
-e2e
+3ee6e9
6664-223669
-326365610--2e378-261646783e e - 3e e5e
e e4e12- e3e1 + 2e1e 5e +
-e 2 e16
22 + e2 e4 e2 - 2e1 3
4
4 16
+ e1 e9e1 0 - e 3e4 e9 - e 4e6 e10
±e2
1 e7e1 ±-2e267 21 - ee 8 e1
elle 3 el-2e 3e4 el 3
-e 3 e 7e1 + 0 e8e
8 1 + e 3 e4-e 2 e3 e 3
-e 2 e 7 e9 - 2e4 e7e9
5
2ele 6 e13 + e 2 e1 4 + ee5e
± e2 e5el3 -
-2e95e13
6
e8 - e2 e6
3 e7e9 + 61e2 e8 e9
- 2 + e20 3e
e26
17
-e3e
- e9ei1
- e 2e-8
e2
± 5e5 el 0 - e 1e 4 e 8 ± 4e 4 e e9 2e e2 e 4 e--3e 3 e4 e + e 75e2 2e
2
+e e16 e9 -ee2 3 e6- 2~e~~e~io+ ± 3e2 ee78+ e2 e4 eseel
2
+ 5e1e3e6e768 4elee6ei0
2e+eeeIe
-3e2 e3e20 - 36ele
4 e2 - +4e22e3e4e
_C2~~~~~~
+ e162e6 e8
+ 5e1e2e6e7e9
+243
-2e6e2 e3e8- 5e2e7e8 +
-- eee
-
e e e - e 2 ee - 2ee4e 7e
2 e4 e9 e1 0 2- 5e1e 5e9e10
+4ele223 e6e 53+ 4e1e2e7 e5 - e 2ee 14 + 3e2~~~i
4e61 3e6e9-223
63+3e3 e26e
1 0-_61364617 + 4217--e3le2 e2 0 + e2e3e-6
1 -+
5
- 6e2 e5e7ell± 4ee 3 e2e11-2e6e4e6e13- 3e e2 2e4e13
-4e5e6 e14
4
2ele4e16 ±
+-2ee 7 e17 e~~~
42632
~~2
_ 6e32ee6ei
2e2e 42e7e12 - 6e-62e 4e72e12
le 6ee 6e7 e 1 ± ee 9
5 6eeo6e 6 + 2e3e2 e5 e7 e 0 -e 2 e2 1
+2e-e3 e4 el4 ± 2eoe8 e5_+ 4e32e
e1 0
+~~
e e3 e% -4e 2 e 3 el 0 + ee 222
7 i - 1'eeloel2
2e3
+2e-e5 e 7 e12± 3e+e4e5 e1 + 5e
3e3e+e8 + 2e3e6 e164 21ee2 ee616
+e-5e
+4ele 2 e2
4e6e
e e9
7 e16+ 3e2e5e14 - 3e 2e 3e5 6
-5e 1 e2 e3 e7e12 + 2e2 e7e16 - 3e26e
± 2ele3e4e13
-9e4 ese6ei0 - 3e 3e2 1 4 - 4elee 8 e12 + 2e2e3e8 e12
2
4
±2e2e4e5e14+ 3ee 6e8 e0 ± 2e1e86 ± ee 23 + e ee 2
e
e
3e
e - e63e742 3e2 4 e6 e9 + 6e1e5e61e 3- 3e 3 6 e 7 e9 - 1 3 9e12
+e2e3~~~~~~
+3ele2eloel2-3ee2eeio ± e4e1 3 - 3e-e8e9-2e2ee 8e9
+3e265e8e1 0- 6e 2e5e8el60± 3e1e 4e6 e + 2ele2 4e16 - 4e2e6 e7 e8
-5e3ee2 + 2e2e 4 e5- 4elel0el4 + 2e3-e8 e14 3e2e9e1 4 2e2 e9 e1 4
-4e 2 ee
6
e9
-4e5e 9 e1
-
6ele 3e81e 3
3e6 el2
4e2e7 el1 2 - 3ele 4e5 e 7 68 - 2e 4 e6 e7 e8 - 3e2e
2
2e6 611 ± e2 e4el69- 3e2e 3 e4 e1 6
6 2e3 e8 e2
2elee e2-eel9
+4-216 66131 4- 6~26l+3
681-- 2e6161
90
-
3e ee2
4-312C612
-4ele 3e 5 82+ 6e26e611 -3e 2 e6e1 1 +
17 - 2ele2 e8 e1 4 - 2 e3e 4e 16
+6ele 2e 5 e 7 + 2e3 e 5 e8 e9 + ele2 e4 e 5 e13 + 2ele3e 21 + 3e2e10 e11
-3e 1 e 2 e 5 e 8 e 9
3e 3 e6 e 2 +
22
2
+
12_21268
e 3 e 4 e6 e12 - 262e 5 e6 e1 2
5
221
0
+2e 2e5e6 e 2 ± 3+5e17+
2eee7 -2ee 2 el e 8 -5e 2 e3ee 0
5
-ele, 4 e 5 e,10 + 2e l e l l e1 3 + e7 e + ele42e5ell - 3e2e 7 e,1 4 - ee 8e 14
-e ee 12
-
2
2
81 7e
2e3e7e 9 + 5ele3 e4 e8 e9 + 2e1e 1 - e6 e9 e1 0 - 4e4 e 10 e11
-
+e3e
1 - 3ele4 e8ell .- 4e3e7el5 23e 9 - 4e1 e 9 + e25 -8617
-3e61e 9 e12 + 2e 4 e9 e1 2 -3ele
e11 1 2 + 3ele 2 e 4 e6 e1 2 - e2 e4 e5 e6 e 8
l2
e2
3 e 4 e7e1 0 - 3e1e 36e8 e
1 o - el 24- 2e 3 e 4e 7e8 - 2e2e4e 8e1 1
-4e 2 e1 0 e1 3 + 5e 2 e3 e7 e 3 - 1e 4 e 7e1 3 -e ee 1 4 -2ele 2 e 5e616
+ 2 e e5 e8 ell + 3 4 5 e 0 - 4e3 e4e5 e 9 + 2ele 3e5 e1 6 22e 4e1 4
-3ee
o2
.
2
2
2
-2e:[e
5 e1 3 + e 6e1 3 + e 3 1e 9 - e9 e 16 + 5e 4 e5 e 8 - le 2e3e1 9
+3e:e 2 e8 1e 3 - 4e7e8 e10 + e4 21 -e-le 2e 3 e 4 e1 5 -7el 1e 3e 4e6 e11
+5ee 5e1 2 e6 e 9 - 3e 2e 3e 5e 1 4 - e2ee 1 3 - 42e 3e 7 e11
-6e]le2e6 e16
ee2e
' 9 el- 3e 1e2ee
4
± 2 2l51 5
4e6 e15
-e3e 7e1 5 + 2e e8 1e 5 - e2e8 el5 - 4e1 e9 e 15 - 2e 3e9 e 11
+4e]e 3 e5 e 7e 9 - 2 e 3e 9 e1 1 - le
1 2 e4 e 18 + 3 2e23e5e1 2 + ee3e
2
+
±
2
1 2 1 ±2
-2e,e 3 e6 e 7 - 242e 4 e5 e12 - e12613
+ e26e
3 e5 e1 3 + 2 e4e6 e 7
+3e 1,e 4 e6 e614 + 2e 2 e7 e 14 + e 6 1e1 - e7 e18 + 5 2 e3 e4 e7 e9
17
+e 3ee 6 e8 + 3e3e9 e1 0 + 4e2e 4e25e9 + 5ele 4 e9 e1 1 + 2e2 e3e 6 e1 1
-e3e 6 e310 + ele 2 e4 e8 e 10
2le 2e 2 2 + 2ele2e 6 e1 4 - 3e2e3e6 e 14
-e 2e 3e6 e14 + 3e2e10e13 + 2e2e
17
- 2e2e 5e13 + 2e2e3e5e7e8
+3e1 e 2 e 3e 8 e11 - e6 e 19 + 4ele 2 e3 e6 e1 3 - 3ee626e12 + ele 2 e65e5
2
~
+3e'e9 3e 5e1 5, 3e2 e2e
7 e10 + 3eee 10 + 4ele2 e 4 e7e1 1 + 2e3e,2
+2e2ellel2 - 2e3 e5 el 7 + ele22 - 2e6 e1 3 - 8e 2e 5e6 e1 1 + 5e 10 e1 5
22 + e e e8
+2e
e4el + 2e 2~~~~~~~~~~
e e5e12e
ell8eII
- 4e
2e
3 4+
3el 03e
2 7 4e8e9
±e'e
e8e3ee
e 3 -3e
+3 e5e5 el
3e3 8e10 2+ e5ee3 84 +
5e720 + e2e
-3e
2 e7e 9 - e3el0 e 12 - 32ee
4 7 eC0 + 5 1e5 - 3e 6 el7 - 3ele2 e 1
-3e6e 6 e1 3 -e e 22 + 2e 2 e6 e1 7 + 3 4 e6 e1 1 ± ee 4 1 5 + 2e2 e4e2
15
-e4e2 - 2e 6e 8e 9 + e2 e6 e6
e
+
2e
e
e
+
3e
e
e2
3ee2e
8 9
3 9 13
2 5
7 e1 3
el2e2
3
3c e2
±e
1 ,ee
8 elee28 + 258- ee4e
0
±2-
+ 2e6e7e2 - e3e34el0
+3e 3 C4e 8 el 0 + 3 2 e3 ee 8 - 4ee 4ee 8 - e3el6 -4eee
7 e9
-e 2 e34e15 - 2e2e5e6 e8 + 4e 1 2e6 e8 + 2 4 e7e1 4 + 5le 4e 5 e6 e9
+2e2e 3 e18 + 2 e3 e4 e5 e1 1 - 3e2e 18 - 4e1e 5e26e7 + 2ele 3 e 5 e 6 e1 0
+ele + 22e 2 e4 e 7 - 3e 4 e17 + e e5 e 6 - 5e 3 e5 e7 e 0 - 4e2 e3 e5 e 15
-8el 4 e 56 15 - 3e 2e6 e 15 + 2e 3 e 4el 8 ee2 21 + 2e2e - e4e7
-2e 2 e4 e6 el3 + 32e 5e18 - e3e
4 e12 - 4e5 e7 el 3 + 3 5 e6 e2- 2ele3 e3
+2e2e 4 e1 9 + 2ele2e 20 + 2e2e 3 e20 - 2e3e 9 e1 3 - e6e3 e1 9 -4e 2 e5 e1 8
2 2
+le2e
- 4ele 5 e1 9 - 3ee 3 e4e 5 e1 2 + 3e 2e 6 e 7e1 0 + 2e 4 e 5e 7e 9
-3e2e 6e 7 el 0 - 3e1 2e3 e17 - 4e2e5e 7e9
-el32el0-
2e2 e 4e 5 e
2
91
2ele 6 e 18
92
Table of Pm,6 for m = 4,5
m
Pm,6
4
2e1 e 3 e8 e12 ee +9e1
e4 e6el4 27-22e6el4
+ 3el5
+2e5e 8e1 1 + 2e4 e 9e11 + e3 e 5 e6 e1 0 - e-el0- 3ee 5e6 e1 2 - 2e3e10 e11
-2e 3 e7 e 4 - 2 2e 5e 1 7 + ele 3 e6 e1 4 - e2e21 + 2e 2 e 8 e1 4 + e 4 e5 e 7 e8
-e 12e 8 e1 4 3e3 e6 e 7 e 8 + e 3e5e6 2- 3ee 6 e1 7 - 3e 4e 5e 6 e 9
+2e 1 e 7 e1
6
- 2ee 9 e14 + 2e162 e3e1 8 + e1e7e, + 2e3 e4 e17 + 3e2e5e1 7
-ee 4el8+ 2e2e6e8 - 2e2e3e8ell + e2e4e 6 + 2e3e5e16 + ele4e8e
+2e 1 e 2e 7 e1 4 - 2e 2e 4e 5e 3 + 2e2 e3 e5 e 14 - 2e l e4 e 5e1 4 - 4e2e6e10
e-e 0e2 2 - 2e22e5 e8 e9 - 3ele 6 e 8 e 9 -2e 1e2 e 4e17
-3
+e3~2
31e,2-2lCe41
+e 2 C7 13 - ele 3 e 9 e6
11 - e
3 e 4e614 + e 2 e2e1 4 + 2e 6 e18 -- 412
3
~
~~2 2 2
2
+e2e2 0 ++e2ele
21 + ele 2 e1 0 ell
3 8 10 - e8
4 e8 - ele 7 e9 + ele 13
-e 1 e 5 e 7e611 - e3 e5e 7e9 + e2 e6 e7e 9 + 2le 4 e 7 e12 + e2 e 3e 9e6
10
+3eel 2 - 3 5 e6 e1 3 + 2e 3 e8 e1 3 + e61e
+
e
e
e
2le
23
2 6 16
2 e9 e 12
-3ee
2e 5 e9 e 1 0 + 2e 4 e 7 e 13 + 3e2e 9 e1 3 -2e 2e 9 e1 3 -2elel 0 e 1 3
+e1 0 e1 4 - 2ele 3e 7 e1 3 - e24 + e2 e 5 e 7 e 10 + 4e 1 e6 e 7 el 0 + 2e 2e 4 e1 8
2_
2
-2ee 3 e4 el5- e52e 14 + ele2el 9 - ele
3 e1 9 - 2ee 3 el9 - ee 2e20
-2e61 e 2 e 8 e 3 + e 3 e3 - 2ele 2 e 21 - 2el1 e 3e5 15 - 2e 2 e3 e7 e1 2 - 2e 1 e 4e 9 e 1 0
1 6 -
e2e2 0
-
+e3 '4el
1 3 + ele 2 e6 e15 + ele4e1 5 - 3ee
- ele22 + e4e5 e11
16
-e 3e5232
e11 + e7 e6
1 7 + 2e 3e 4e 5e1 2 + e5 e 19 + e5e9 + 3e3 e669 + e5e 7 el2
+2
e 2e65 e1 5 - 2e2e 4 e7 e11 + 2e7e89 - e 2e 322eee
e616 + e2 e3 e6 e1 3 - e2el8
-2e2e 7e1 5 + 2e1e8e15 + e1e3e60+ e3e4e8e9 + e-e7e9 - e2ee 8
+2e6e3 e 4e1 6 e+2 e22 -ee 5 e1 3 + 3eee 1 3 - 3e 3e4 e6 e11 + 4e2 e5e6 e11
32l- 2e 1e 5e 18 + e2 e4 6e e1 2 - 2e 1 e2e5 el1 6 + ee 11 e1 2 + 3e21e 0 e1 2
-e 2 e5e1 2 + e6 e8 e1 0 + e9 e15 + 3e6e7 1e1
3e6 e7 e1 1 + 3e2 e4 e8 e1 0
-
-2e 1 e 5e8 e10 + 2e1 e4 e1 9+ 2e 1e 3e20
5
-e 1 e7 e8 e1 4 + 4e6e
- 9 e1 5
3e1 e2 e6e 11 ± 2e 3 e6 e1 0 e1
-
5e 1e 2e6 e1 0 e1 1
-e7a 9 -e 1e 2e 3e 2 4 - e6e 9 e 18 - e1e3 e 4 e5e1 7 5e1 26e
e5 e1 0 e1 2
-5e3e 5e1 0 e1 2 - 3e6ee 1 9 ± 3e 3 e9 e1 8 - 3e5e8 e1 7 + e8 e21 - e4e7 e11
+2e'~eoe 8 + 3ee 6 e - 4e2 e10 e 8 + 2e 3 e5 e - ege eo ± 3e6 e7 e+7
+3e6e11 e1 8 +
e6 e7 e8 e9+ 3e2 e1 2-e1 6 e4e 10 ± 2e6e5el 19 - 2ee 10 e 14
22±-2
-e
3
e 17- e 2e
1 7 e6
1 0 e 1 1 + 5e 2 e 7 e1
-2e5 e7 el 8 ± ee2 ± 3eee
-4e4 ee6 el0 - 6e4 e5e 01e
6e
5
0e
11
-2e4e
8 e1 0
-7e
2 e4e 5 e 7 e6 12
+ 2e1 e2e9 e1 6 + e3 e5 e6e8
elee + 2e6e4
+1 l
1+3ee
6e7el
+e 2 e 7 - e4 e2 6 - e4 e18 + 4e1 e3 e 5e21 + 2e2e5 e21 - 2e 3e4e9 e10
-4e 3e6 e 8 e13 + 2e2e7e 8 e1 3 ± 2e8 e9 e1 3 -2ee 6 e18 ± 2e64e
7e 15
±2el 3 e6 e7 e 3 -4e3 e4 e8 e 2 ± 2e 2 e4 ee 1 4 -4e 2 e6 e 1 ± e2ele22
7-2e
-7e
e e e -8e e e e ± 4e e e e
3 e 4 e e8 _ 6e1e e e
7
-- 23646762,
5 91 5
4 6 8 12
-- 6617s691 -- 7646686126
--
93
5 6 9 10
86s4+
696o 4
3 4 1 0 13
6C9
346106,14
+2e2e 3 e7 el8 + 2e2le4 e5 e19 -4ee 6 e8 e 4 + e2 e6 e8e14 ±+2e 3 1 0 e1 4
+ ee636466617
6j--612636-7617
1
-46172 12451
±
3e4e6e1 7
+5e 3 e+56364668610
4 e5 e 8el0 - 4ele7 ell - lee2 e4 e5 e18 -6ee 2 e3e7e17
-
+2e2e 4 e7e17 + 26 3 e4 e6 e8 e9 - 2e6e22 - 4e4 e5e6 el5 + 12e5el0oel5
2
~~~~~~~~~2
2
22
4
+3e3 ee 6 e13 + ele 3 e6e17- 3 2 e5el8 - 7ele2ellel6 + ele7e 1
1282 e8 el9
381
- C~-4ele2e4 e7el-ee
+ 416,264e67e6
6611613 +
8 Ci6 - ele
+2124e66617
+2ele
2 e4 e8el7-- eelle13
el
e
-4e 3 e 4 e 5 e2 + 2e 2 e+e7 e13 + 4e 2e 4 e6 e8 e10 + 2e 2 e3 e6 8
-22e
5
e 7 el1 4 + e30 + 2e1 2e 4 e 21 - 43e 5e6 el1 3 + 4e 10 e 20 + e 2 e 3 e 5 e 9 e1 1
+5e 4 e 5 e 9 el12 + 2e 5e8e9 - 3e 3e 7 e2 0
ee
+4ee
1 o3 -3
1 65 e8 e6 - 3e1
~~~,
2~ ~ e2262
+ 6e2 e
e16
e
+ 3e26
13
+6e25e20
4 6
le2
~
4 7
2
12
-
8 e03ee
4 e2 2
3ee 8 e0e1 + 8e5 66e 7e12
ee12,5~l
1 2e5 e9 e1 3
2
-2e2e6e4
22
6e
1
e2e
-e 3e 5e6 e 7e9 + 2ee7e615- ee 6e 7e10 + e6 e 7 e10 - e2e3 e4 e21
+2e2e 2e 3 e23 - 4e2e8 e9 ell + 2e2e4 e1 0 e1 4 - 2 e8 e 9 e11 + e23e5e 6 e1 2
e 5 e 2 2 + 2ele 8 e21 + 2e l0 - e3e4e15
2e 2 e 7e21 3e
3e2e 7 e21
_2e3e
2
2
-3e 3e 4e 6 e7el10 + 4e 4e6e14 + e 4 e2 2 + e 2e 4 e 8 e14 - 4ee 4 e6 e 8
2
+5e3le e - 3ele 2e 3e2 2 + 3e 3 e5 e9 e1 3 - 6e 2 e4 e 7e1 7 - 4e 4 e8e 9
6
21
e2
4ee 222e10e1e6
± 3e1e2 e e-4eee2
-3e1e4e5 e20 ± 3e6ee3
- 2e4e 5 e8 9
e15
e
2e2le
+
e
e
e
4e
+
e
e
9
4
l 2 5 22
+26 2e 5e6 e1 7 - 32e 3 7 1 6
4
2
2
~~222
2
ee
2e
- -3ee 7 e8 ell + e3e8el2 + 3 13 + ele26
-46 e2 e7e1 + 7 e6
3 5
2
2
e~2
4
+2e1le2e6 e 8e13 + 3ele 2 e4 ellel12+ e4e1 4 +-2e2 e3 ee17 + e2e6 e 0
-3e 3e 5 e22 - 32e 9 e17 + 2e 4e 9 e17 + 2e2e2 e 9e1 7 + 3ee5e 7 el0
3e 2e 3e6 e 19 + 3ele 3e 6 e2 0 - 22 e3 e6 e1 7 + e4e25e12
-e3341-4e26e22- 3e 2e 3e 2 5 - 3 2 3e25e1 5 + 2e6 5 + 5e 2e 4e610 e 14
-6le 3e6e 9 e11 + 2ele 3 e5e6 e15 - 4e2 e4e 6e ± 3e2 e6 e2 2 -e-e 5 e 19
2e6 e7e 13
-3e 7e 0 el 3 + 2e21lle15 + 42e 7 e17 - ele 2 e15 -e8 22-2
+ 2e 4e 7 e19 + 5ele 5 e6 e29 + 3ele 3 e5eo0e
-2e2e 3 e9 e16 + 2ele 2e321
3
1
+2ele 7 e2 2 - e2e3e2 1 +3e2e 3 e 9 e1 4 + 2e l e3e 7 e1 6 - 4 le4e8e16
20
+2ele2 e 8el8 - 2e2e 3e 5 e20 - e2 ee 13 - 4e2 e4 e2e10 + el2e4e
e
2
ep2
+ 2el e13 e16
e el14
e2e
4 el8 + 3e2 e4e8 e12+ 4e2e 365e68e12- ele 3 6
- 6ele5el10el4
e7e62
e
e
+
2 3
-6el62 e 3 e4e20 - 4e 3 e5e7 el5 + 5e4 e5 e8 e13
+ 4ee527e10- ee37e8
3e27e12
-4ee 2 8e10- 3e25e8ell - 3ele
2
e22
62
-4e2le 6 e7e15 - 6ee 3e5e 8e 13 + e7 e + e2e22 - e1 e4 e11 e13
-e3e 8 e1 9 + e2 e3620 - el e4e 7e 18 + 4e 2 e3 e 7 e18 + 6e l e 3e4 e9 e1 3
-3ele2e7 els + e4e6 e7e13 + 2e6e4e20 + e2e4e6 el8 + 2ee 5e6 e18
-3eeel4 + e2e2e +-4e1 e2 e3 e8 e6 - 2e4 elle3 - ele3e4eloel2
-3e 5 e 7 e - 2e4 e6 e2 ± 02 1 ± e2e13 -- e 1ee 20 + 2e3 e8 e19
+2e2e 9 e1 9 + 2e 2e9 e1 9 ± 656se 8 e -3e 2 e6e 8 e 14 - 8e1e22 e5 e6 e1 6
e 4e 5e6 e1 2 e4e 3
-e 1e 5 e ±- 4e2 e4 e5 e19 2e3 e4 e2e1 3 + e3-+
+2e 3 ee6 -4e2e 12 e16 -4e6e 2 e1e1 5 ± 2e2 e6 e7 e5 - e 3 2 7
e e6613± 2ele3 eel8 ± 3ele2ele14
-2e2e
3e
2 8 e17- 3e2e 4 5
+2
±ee4e61-2ele6eel2
2- 2
-e 3e 467e16
-56C17
+ 2e2e4e5e15- 3e3ee1
+ 2ele25e120
2e-
-ee
3
e
2 623
±
22
2
2 ee 99
94
3e16561e63
-3ele
2ee
13
+ ele 0 e12 + 4e 2 e7 e8 e1 3 - 2e 5e6
1 2 e1 3 - 3eee
13
'2
2
-[46;ii68¢10
-+±661636e8612±e
2e~e3 e4e 9 -3e86263667612-323e669610
2e3 e10 e12
3 e7 e20 ± 2el~e
8e20 + ele6e7e9
-2ee 5e 7 e1 6 - 3elel 0 e1 9 - 3e3e 10 eo17 + e-e 24 + 6e 1e 4 e6 e 7e 12
+5e,e 2 el 2 el5 + 2e2e 3 e5 e 2 0 - 4e2 e 2 el 4 + 5e2ee 1 6 - e2 e62e16
+2e 1e 5 6 e 8 e1 0 - 2e 3e 4e 5 e 7 e1 1 - e33e
5e 16 + 5e 2e 3 e7 e8e10
+2e:1e 4 e6 e 9 e10- 3e 1ee 12 613 + 2e2e 4 e 9e1 3 - el1e 3 e 7e 19
-6e2,e 3 e 5 e 20 - 32e 7 e 19 + 2e2e 3e 23 + 4e 4 e7 e 9 e1 0 - 6ele 2e 4 e9 e 14
+4e, 5e 2 5 + 4e 2 e3 e4 e7 el
1 4 + 3e 2 e5 e7 e1 6 - 3ee4e 7 e1 4 - 4e2e 3 ele 1 4
-5e,,e 5e1 l0 el
1 3 + 2ele 6 e10 eo1 3 + 3e 4 ee21 0 + 3le 2 e 3 e611e 13 + 3e 2 e 3 e 5 e 7 e1 3
-4ee 7 e8e9 + 3e l ee 1l0el - 2e8 l0 e12 - 5ee 4 e2e13 + 3e4e12 el4
+ele, 3 e 7e9 e1 0 - 3ele 4e1 5 -321
- 3e 5e9 e 12 + 3ele4 e5e9 e11
p~~2e2
2 2
6269
-el,'2
- 6e31 e 4 e 5e 20 + 2e3e11
+ 2e4e6e
1 0 + e 4 e5e 7 e 9
+e 2 e5e9 - 2e6 e3 + e l e 4 e 5e7 e1 3 - eel 3e 14 - 2e2e4e 10 e1 2
2 2
-3e,!e699el3 - ele7e96l3 + e26e5e16
-3e465616 22e3e8e9
-5e,e 3 e 4 e 8 el1 3 - 3e 36 2 3
e 2e 3e 5e6 1+4
2e 3 e1 2 e 13
1 e 4e 23
+2e 2,e 5 e1 1e 12 + 5elee 6 e6
2e
e
e
ele4e6
2ee
13
2 4
17
3e 5e 9e 12
3
e2
~~2
+2e 3 el 3e 14 + 2e4e - 3e 2 ee 9 1e 1 - e2e 3e61 1e 14 + e4e 6 e8 + 2e 7 e 9 e14
-4e 1 e6 e 7 e1 6 - 4e3e6 e 8 el 0 + 32e 1 0 eo16 - 3 2e 4e 5 e 8 e1 1 + e2e4e11
22
22
+e2 24 + 3ele9el1 + 3ele 4 ele
-e 2 e 5e 6 e 1 1 -
2
-
0
-- 4e 2ee5
14
+ e~2C2
2 2
+e~~__2
e2
+ e6e5 18 + elel 4 + 5e 2e 3 e 2 e13
2e 2 e66e + 2ele
e
4e 5e6 e 1 4
-5ele 5 e6e 12 + 6e52e0 - 4e 6e618 + 3 3e8 e9el 0 - 3e 4e 7 9
+3eaee 8 e9 + 2 3 e4 e2 3 + 4 2 e6e7 e 9 - 56e 7 e11 + 3 4 e 5e 7e14
-3e1 e 2 e 4 e 0 eo
1 3 + 3ele5 e 7 el
1 7 + 3le 3 e 7 e 8e1 1 + 4e3e8 e1 3
-3e 2e32e4e 18 + 5 2 e4 e 5 e9 e10 + 6ele6 e8 e1 5 + 2 2 ee 5 1 7 - 1 e2e 4e 23
+e3e1 8 + 2e2e3 e5 e 8 + 4e 2 e5e 9 e1 4 + 2e
6 e 9 e 14 + 4e 2e5e 7 e1 1
-ele 2 e 5e8 e 1 4 + 2e2e 3 e25 - 3ele 4e2 5 + 6ele 2 e6 e7 e6
1 4 - ele 3 e 4e 7e1 5
-3ee 2 e226 - 4ele 3 e4 e2 1 - 3e 3e6 e 7 e1 4 - 3ele 4e1 2el 3 + e163 e 8e2
3
2
2
-ele 2 7 + e8 e14 + 3e 3 e4 e8 el - 2eC 6e 7 e9 + 6e 4 e6e 9 el - 2e2 e3e 9 el6
+3ele 2e 5 e 7 e15 + 5ele3 e4 e8 e14 - 3ee 4e 5 e 17 + 2ee 5 e 17
+2ee 2 e12 e 4 + 2e2 5 e 9e14 - 3 2 e 5 e23 - 2 5 e6 e19 - e4ee 1
-3e2e 3 elle,12 + 2e2e5 e2 3 + 2ee 7 e,17 + 3e2e3 el0 e13 + 2e 2 e4 e6e 7 e
-3e 2e 3e 4e6 el5 + le2e7e12 + 4 e8 618 + 4ee 15 + 3 1e6 e 23
+2e 3 e4e7 e 12 - 4ee 2 e6 e2 0 + 2e l ee 4 e19 + 3el 1 2 e1 7 -le29
C
2
+
22
+2e2e
2e2e
2e 5 e 0 e1 3
3e1 0 e1 5
8 e1 8 - 3eele 1 3 + e2e3 e6 el6
-5e 3e1 2 el5 - 3e 2 8e20 + e2e 26 - 2e3 e5 e6 e - 3e 9 e20 + 4e3 e4 e5 el8
le2e
- 2e2e6 e8 e 12 + 2ele3 e e 16 + 2e2 e e20 - 4ee 6 el0 e 12
3
ee23e 2"7el - e4e5 e613 - 4ee 5e611e 12 + 3e 5 e 7e 2 -4e 2 e63e
0
+6ele 3 e6 e 20 - 7e5 e7 e8 e 10 + 2ele 7 elo0 e 2 + 2e2 e3e 7e1 5 + 6 2 e6 e10 e12
+2ele 2ee 17 -3e 3 ee 1 7 - ele 2e 13 e14 - 3ee 2e 3e 22 -9e21
-6ele 3 e 8 e 8 - 6e1e 4e10 el 5 -2e e6 e, 2
3
117- 4e2e 11e 17
+e-2ee 2 4
2e 2 e1 1 e1 7 - e2 e 2 8 -3ee323
-2el
1 e 3e 5 e 7 e 1 4 - 3e2e9 e13
-
95
-7e 3 e 4 e9 14 - 3ele 2 eel9 - e6 e 8el 6 - 4e 2 e3e9 13 -37614
e 4 + 4ele 4e9e16
-8e 2e 3e 1o6105 + e6 4 6e14 - 2e 2 ee 6 e 14 - 3e3 4e 5 1
e15
-367 e 8 e 15 + e3e5 e6ell - 5e3 e4e6ell - 2e3 e7 e9 ell + 2e 2 e3 e8
ee
e
+
e21
e
5
-2e6e 8 e14 + 2e6e2 e4e22 - ee 3 e24 - 3e2le2 5
-2e1 5 e7e8e9 - 3e3e4e19 + 2e6e7e9e12+ 4ele2e 7e20- 3e4ee15
+3e 3e6 e21 - 3ee 3 e6eo +±ee 5 e9eo - 3e9 e1eo + 2e2e22
-3e2e 8e15± 3e1e2 e7 e9 e1 - 3e2e3e 4e21 + 4e2e3e6e13
-6ele3 e el5 -3ele 4 15 - 2e e12 e13 e9 e 5 +-5e2e6 e7 e11 - e3 ee1 2
-3ee 9e 0 + 2ee4e24
+-3e
2e5e7e ±3ee4ee9-e 7e23 -2e 2ee 0
+
e e0
-2e 6 e9 e615+ 2e1e 2 e6 e9e12 + 5ele 2 el0el7 + 4e 3 5 6
2e6el
7 e1e12
3e4 e8 e5 +2 3e
56163611
-5e2 e7 e 9 e12 + 2ee 2 e7 e1 9 - e2eee2 19 + 2e36164676861o
± 2ee e21 ± 2e2e4e24 + 3ee 2e6el9
±4e2 e4 e9e5-4e-esel2
-3eie1e 9 e1o + 2e2ee 2 2 + e3e4e 0 -3eie 4 e7 eseio ± 5eie3e
+e 2ee612- 3e2 e13 e15- 51e 4e6 e8e11- 3e6ese16- 3 1e 5e24
6e 1e2 e 7e 8e12
+5e6el16 + ee 2 8 + 3e26e5 e6el15- ele2 e 3 e5el9 +6e3e46
1-23e4e5e61
2
-
2
-l3961 + 52e4e76e8e9 -3e5e96 1
--226769e0 + 4e2e46e62-e616e64- 462636619-e639615
1 - 3e1ee 9 e1 2 - 2e2e 4e 2 2
-e26 2 4 ± 3e2e6 e20- e1e2e5e2
el0el6
e 3eee + 4 31ee
e2ee17++323492
+ 42387
6469
+3e1e 5 e 22 eleee9
e21 - 3e 4e 5e21
+2ee6e 5 e616- 3e 1e2e 8e17 - 3e2e3ee1 - 8e6e12 e 6
-eiie - 4e6e7e13 - 4e 2 e 6 ee8 + 3ele4eei1 + 2e2el 3e15
-3e2 e5 e6 e8 e9 + 3ele1ee112 - e 14e16 + 3e6e5 e6e1 - e6e2e25
- 1 6 2e 1e 3e 9e 17 + 6ee 2 e3 e6 e1s
-6e 4e6 e2 0 - 4e1 e3 e4e6 e16
e15-3ele2e 5 e8-ee7e2
2 4 e 539e6e
e9 ei-~
5 e6 e7e61 + 2e62e6
4 3e~e
6e2 e5 e8e15 + 6e 3 e5 e6 e16 - 3e 4el0el6
2eee25 +5e2e5e6e7e + eee 9e 4 -2e 6e24 3ee 4e7e
+26162625e1
+ 56C285
+ 3712e,56015 6662e4ee16j2467e61
-4ee
6 e1 4
+ 2e1 e 3e26 + 2e1e2 e27
-
96
Table of P4 ,n,for n = 7,8
n
P4,n
7
-2ee 1-26,16
e~
2ee ~
e 2e23 - e2e
-2e 3 e12e
13 - ~
9 e1ll~+
+ ~13
626123e
2 3891
3
e52~-9I
e9 e1 4-2ee
-+
4 e2 3
+2ee 3 e2 3 -3e 5 e7 e1 6 + e2e6 e1 6 + e2e 3e 23 - 2 e6 e8 e1 2 + e 361 1e 14
2
-ele 2 e 2 3 - e2 e42 e 8 - 2ee 3e 2 4 - 22e8 e 16 +e e6e2e 24 + 2ele 2e2 5
+6e4Ie 8e1 6 + 2e1 e2 e 5e2 0 + 3e4e2 4 + ee 1 0 e1 6 + 2e1 e 5 e6 e 16 + e 2e 3 e1 1 e12
-2e:1e 4 e1 e 12 - 3e1 e7e 8 e 12 - 2e2 e6 e 9 e1 1 - 2elelle 1 6 + e 7 e10 e11
+3e 2 e6 el20- 2e 2 e1 0 e16 + e 2e 7e6
1 9 + el1 e 2 e 1 2 e 13 - 3e 2 e 7 e 9 e 10
+2e [e 8 e 9elo0 + 2e 3 e 4 e6 e1 5 + e e 22 + e 4 e6 e 7 e1 1 - e3e e 1 7 -3e 4e 5e 7e 1 2
e5e8 - 3e4 e 7e8 e9 + e1 e7 e 9 e11 - 2e1 e2 10 e 15 - 2e4 e9 el1 5 - 3e1 5e 17
+3e,"!6 e18 + ee
16
- 2e4 e6 e18 + e
21
e21 4 - e62
15
- ee313
+4e, 3e 7 e18 + 2e 2 e3 e 4 e1 9 - e6e22-3ee
2
7 e 2 0 - 2e2 e8 e 18 -23
C2
2
2
+e2e8e1 8 + e5 e 8 e1 0 + e3 e8 e1 4 - 61627- 2 2e 5 e1 0 e11 - 2e2 e6e 20
+2e 1],e9 e1 8 - e2 e 4 e1 0 e1 2 - 6 e8 e 14 + e2e1 0 e1 2 + e 2 e4 e 7 e 15 - 2e6 e1 0 e 12
+ I 2
-e_22e3 e 2 +
ee 3 e21 - 2 3e 4e 21 - 32e2e 5 e2 1 - e 8e120 + e3 e4 e7 e14
22 -
+ e2 e 3e 20 + 2e 5 e1 0e1 3 - e22e1 1 e 13
e62e2 6 + e636e
1 0 - 2e2 e 4 e6 e 16 + ee 4 e1 0 e 13 + 2e 1 e 3 e 5 e 9 - 31e 9 e 17
-el1315+ 2ee 2 e 8 e1 7 - e 3e 2 5 - e 9e 19 + e 4 e83- 2e3 e5 e8 e 12 + e 2e 3 e 7e 16
+4e 2 e 7e8 e1 1 - 3eje 6 e 7e 14 - e 3 e122 + e25e
6 e12 + 628 + 3 4 e7e 1 0
2
2
2
+-e4e 25 e14 + 2e 2e 5 e2 1 -e
3 6 e 13 + 3e4 e 2 + 2e 4 e 5e6 e1 3 - 2e3e7 e-15
+2e 1e 4 e 5 e18 + 2e 2 e 4 e 5 e1 7 - 2e 2 e 4e 2 2 + 2le 5 e 22 - 261 e 4e 8 e 1 5
-6469611 + 4 6 e 7 e1 5 - 11 617 -- 3e+1 e6 19
2e
5
9
+2e5 e6 e 1 7 + 3e l e2e 13 + 3e 3 e5 e 9e,11 2e 3e6 e19 + e2e 961 3 + 2e6 e9 e13
+2e2,e7 e1 1 l - 2ee 2 e11 e1 4 - 2e4 e 5e19 + e262 16 - 2e1 e3 el 0 e 14
+2ee 5 e1 0 e1 2 - 2e5e11e 12 + 3e4e 20 - 2ee 3e 8 e1 6 - 2e3e 5e 20 - 2e3 6 e9 e10
-2e 2 e5 e6 e1 5 - 2le 3 e6 e1 8 -3e 7e8 e1 3 - e5 e2 3 + 382e1 2 e2e2 2
+2e 2e 5e 8 e 13 + ele6 e 8 e61 3
62le64622- 2e3 e4 e9 e 12 + e 3 e 4 e10 e1 1
-2624 - 3 4 e7 e1 7 -2e
3 e8 e 1 7
-ele6e11 - 2e 3e4e2 0 - ele9 - e2e12e14 + 2ele3e9 e15 + 2e1e66 21
+ee,5el
7
+ 2e 2 e 3 e 8 e 15 + ee1 61 3 e 14 - 3e 5 e6 e 7e1 0 + 2ee 2e 4e 2
- e10e18 - e 35e 0 + + 2 +2e3e7ed + e2 e8 e2
~+2eV
-2ele 6 e 0 e1 1 + 3e1 2 e1 6 - 2e1e 2 e6 e 19 - 2e 3e5 e6 e 14 + 4 3 e6 e 7 e12
-e 42e5 e 15 + e3 e2 15 6625e3- 2ele 2 e 3e 22 - 2 3e4e 5e 16 + 3e1 e 3 elle 13
-2e 2 C3e 6 e1 7
2ee 4e 6 e17 - 3e1e 2e 7 e1 8 - e6 1 6 -- 2e 2e 3e10 e 3
-2e 5e 8 e1 5 - 2 2 e 3e9e1 4
e2C +
2
-e 2 e8 e 0
6611615
2
-2e 3 e4 e 8 el3 + e 3 e6 e8 e1 1
+ele 5 e 7 e1 5 - 2e2 4e8 e 14
2 1e 4e 9 e14 + e2 e6
5 e9 e12 + 2e2 e 11e15
e2el_2
3e10 e 15 - 2ee 12e 15 - ee5 6 e11 -e42e 6 e 14
- 2e4 e 5 e8 e1 1 -e 4el 0 el 4 + 2ele 6 e 9eI 2
- 64e6e8e10o + 3e2 e6e14 - e3e2e9
e 2e 4 e2 1
2e 2 e 9 e1 7 - ele2el9
2e 1e 10oe 17
e 3 e 7 e 8e 10o
+26 26 4 e 9 e1 3 - 5le616569613 + 2e616568614 + ele616562- e2e
5
+3e
8
e
20
2
ee - 2666,5,1
-2 62-2656e19
++36~~26
3 210614 ±e+
34e18
l620
223518
+e616367617 + 2e2e7e619+ 6e2elo0- e2e26 +- 636567613 - 36266676e13
-41e2
97
+3ee 8 e1 2 -e 2e4e2 0 - 4e3e2ell - 2ele8 e 19 + e4e5e9 e1 0 - 2e3 e9 e16
8
ee
11e 13
+ 3e 3e7e2 1 + 2e1 e7 e2 4
-
e2e 12 e1 4 + e3 e1 3 el6 + 3e5e9e 13
~~~~~~~~~22
- 2e6 e 9 e17 + e4 e
9e1 7 - 1011
5 ellel2 + e5e7e8e1 2 - e26e
e,~~~~~~~~~~~~213615
3 e1 5
+2e + e8ellel3 ± 2e6 e1 2e14 + 2e1 e2 e1 1 e1 8 ± ee22e
-2e 2 e4 e26 - e2el4e16
+3e 3ee15
+-2e9 elle12 + 2ele 2 e6 23 - e3 e5 elle13 - e2ele19 - 2e6elle15 + 3e2 e1 4el6
-2ele4 6 e21 - 2e3e10 e1 9 - 4 4 e 12 - 2e5e9 e 18 - 2 3e6 e7e616
1 2 e1 4 - 3 2 e 7e 8 e1 5
- 2e2 e1 1 6e
1 9 - 2e3 ele 1 8 - 2e1 e2 e2 9 + 3e 2 e4 e6
- e2e5e2 0 + 2e l e12 e6
1 9 - 2e3e6 e 23 - 2e3 e 4 e6 19 - e 1e9 e11
+1615616
+e-5 e8e1 9
+2e 4 elOel8 -elel 2 e18 + e2 e 7e 21 - 2ele 5 e2 6 + e3 e4 e8 e1 7
e 7 e 23
+-2ele 6 elle14 + 2e 2 el2el8 - 2e 2e4 e9 e17- 2e5 e10 el7- 21
-2e 3 e 7 el0oel2- 2e 5e6 e21 - 2e 3 e6 elle 12 - ele 4e 26 - e2 e4 e23 + e3le2 9
+e 6 e 26 + 2e 2 e 5 e 6 e1 9 + 2e 2 e6 ele 3 - 2el1 e 5 e1 2 e1 4 - 2ele 1 3e 18 - 2e2 e 7e2 3
-+ele6e1 9 + 2ele 7 e1 0 614 + e2 e 11e1 7 + 3e l e9e 1 3 + 2e 3 e 4 e 9 e1 6
2el1 e2 e13 e16 + 6e165 e9 e17
-+e20 e1 2 - 2e3 e5 e9 e15 --
+ele 8 e23 - 3e2e 24 - 2e 2e 7 el6le12 + 4e 3e 8 e21 - 3ele 2 e 8 e 21 - 2e4e6 e 9 e13
11 13
+2e 4 e1e17 +-2ele 2 e 3e2 6 + el 3 e19 + 2e 4 e 5e 7 e16 + 2e 3e1 2e1 7 - 2ele7 ee
e6e
'
e
e
e
ele
+
23
1
2 14 15
-2e 5 el 3e1 4 - 3ee 10 e61 8 - 2e2 e1 3 el 7
+ 1le26
e
e
-2e
elel4
e
e
+
e17
3e21el
4e25
+
2
1
27
-2e 2e 3e 4e23
2 5
3
+2e 2 e 5 e7 e8 + 2e 3 e 4 e25 + e 2 e 28 - e 6 - 2elel 4e 17 + 2e 4e 8e2l0
e - 2e7eloelh+ 2e7e12e13
2ele3-eloe8-ee+ele
8 ellel2
462166 -27610e15 + 2676-12613
- 212 1424-4 -4e
+ +263108
-[[68162
-ee 3 e 2 7 + e2e 9 e19 + e 4 e 4 -2e 4 e 5 e 9e14- 3ele 6 e 8 e 1 7- 2e 2e 4e7e 19
1 0 el1 4 - 2e 5e 6 e9 e1 2 + e 2 ee62 2
-3e4e22 - e3 e7 e9 el1 3 + e9 e2 3 + 2e 3 e 5e6
A-ee2 0 e61 + e26e
4 e 24 + e 14 e1 8 -2e 4 e 5 e 6 e1 7 - 4e 20 + e3e1 7
-e 9 2e14 + e3 e5 e8 e16- 2e 3 e 4 elOel15 + ele 4e612e615+ e2 e 3 e1 3 e1 4
2
2
+ e3 e4 e21
+-2e3 e6 e9e14 - e6 e7 e12 - e5e 6 el 6 + e4e5e8e15 - 262e 3 e2 722
2
+-2ele3 e4e24 - 3e2 e8e9 e61 3 le 6 e24 + e3e6 e1 0 e1 3 + e3 e6 e17
- e2 eel6-e 2e2 2
+ 2ele4e2 7 - e6e20 + 220
+-2e2e 3 elo0e17
-
3e1 1 + 2e 2 e 4 e6 20 - 2e 8 e 24 - 2ele 4e 1 3e 14
-e 4e6e 1 8 + e2e 6 e61
8 +
+
2ele 3e 2 8
-ele 2 e28 - e3e 26 +- e 26 7 e 9 e14 + 2e 2 e 7 el0el3 - 3 e6 e 20
e
3 2 - 2ele 4 el0el7
+-e2 e6 e8 16 - e2 e29 e12 -- e63e14+ 2e 2 e6 e 7 e1 7 + ele 31
+-2e3 e5 e6 el18+ 6ee9e1- e6e2 3 + e6
11e2 1 - 2e 8 e12 - e5 e7 e9 ell
+6e4 eel6 - 3e 1 e8 e9 1 4 + 3ee 1 1 e1 5 + 2e 2 e 5 e1 0 e1 5 - 2ele 5e6 e2 0
- 3e61220
+ 4ele768616
+ e4e5el9
+2elC667618
- 2le,4C9e8
-2 3e3e7e8el4
+ e~e6l
2e2e3e9el8
-2elel0e21
+ 2e465623
+e265e23
-3e 5 e 8 e9 elo + e7e 9 el6 e 5 e2 2 - 3e 2e6el 8 - 2ele 5 e 7 el9 + 2e l e 3 e12 e616
+-ele6 elOel5 + e2 e4 e8 e18 + e15 e17 +- e2 e30 + e5 e2 7 + e2 e8 elo0el2
-2ele 3ele 17 + 2e2e3e5e22 + e5 e6 e1 0e11 +-2e4e9 e19 + 2e5e1 2e15
2 + 4ele 4 e8 e1 9 + ele 3e2l4
+-2e4 e7 621 + e10 e 22 - 2ele 4 e 5e 22 - 2ele 9e 10 e61
2
_
2
e 10e 12 A-4e 3e8 e9 el 2
2ele 3 e7 e2 l - 2e 1e 2e 12 e1 7
- e2e5 e 2 1
+-2e26e
8 e12 A eel0e14 + 3e2e 7 e19 +- 3e 4 e 6 elo0el2
A-e3 ege1 0
-
-
22
10
-+3e
5 e6e26 - 12e5 e7e17 6-
2e 3e 5 e2 4
--2e 2 e 5 e9 e66- 2e 3 e 4 e1 1 e14 - 3e266e22-62630
98
2
-e 3e 5e
9 -2e 2e3 67e2o
2e 4 e7 e10ei1
+62e 7 e 13 + 2e 5e7 e 20 + 3e 5 25 - 2e 2 e5 e2 5 - 2e3 e 14e615 + 4e 8e 9e615
+3e e2e 2 1 + 4e6 e 8e 18 + 2e 2 e6 e9 e1 5 e 7 e9 e 15 + 2ele 4 e 7e2 0
-2e1e6 3 e 5e 2 3 - 4 6 e21 + e4 e 8 e 9 e1 1 - 2e 2 e 4 e 5 e 2 - 2e e16 e 2 5
-e2 e625 + 2e 2e 4 e 0 e16 + e 8 e1 0 e14 - 5e2 e6 e 10 614 + 2e4 e6 e22 - 2 4 e 5e 10 e 13
2 +e
23
+e 3e 4e61 2 e 3 - e4 e29e10 + 3ele 5 61
3 ± e2e 3 e2 5 + e4e 7el
1 7 + e 5e9
+e 5e 7 e 0 -3e 3e 8 e1 o0e11 - 26e7el9-e 3 e-10 e 16 - 2e 9 e1 0 e 13 - 2 3e 7e 22
+e 7e 25 - 2e 2e 5e1 2 e1 3 - e3e9 e11 + 3e 6 e1 0 e 16 - 3le 5e8 e1 8 - 3e 5e6 e 8e61
3
+e 4 6 e 8 e1 4 ± ee 9 e15 + 2 8 20 + 2e 2 e9 e6
e6
7e
e
1 0 11
4 8 20
-ee
3 e 3eI 5
- 2ele3e9 e 19 + e 2 e8 e22 - 2e2 e 3 e12 e1 5 - 2e3e 4e 7 e1 8
-2e 2 e 4 e 1 e1 5 - 2ee 2 e5 e2 4 + e3e23 - 2e4 e6 e 7 el15 - 3e6 e7 e8 e1 1
-2ee 6 e 2 e 3 -e 72e618 + 2e 3 e9 e 20 - 2ele 9 e 22 - el e32e2 5
32ee 9 e 2 1
2
2
-2e=,e9 e2 l elele20 -e 5 e 7 eB3 + 2e 2 e 0 e 20 -e 2ee27 e 16 + 2ee 2 e7 e2 2
+e 5e:1 1 e1 6 e 2e 32e24 + 2e l e l l e20 - 2 2e 3 ele 6 + e6 e7 e9 e10
-ee 5e611 6e15 - 2e26 + ele3 e8e20 - 2le 6 e9e1 6 - 2 7ellel4
-2ele 5e 10 e 16 + 2e1 e3 e6 e22 -e 5 e 7eI 52 - 2 330 + e3 e5e~~~~~~2
122
e2
+2ele 4 ellel67e7 - 2ee2 e9 e2 0 + 2e 4 el3 el5 - 3e 4e28 + 2 2 e6 e 24
+e 4e 7 e 9 e1 2 + e 3 e2 9 + 2ele 2 el 0 el9 - 5 4e 1 2 e 16 + 2e 5e6 e 7e1 4 + 3e2 e e1 4
62
22
+e 2e6 e2 2 + 4 4 e7 e8 e13 -e 4e 7 e 14 + ele7 e2 2 + e 7e8 e1 7
-3e 2e3e 8 e 19 + 2e 2 e 3e6 e2
-
-
-
~
99
References
[Ati67] M.F. Atiyah. K-theory; lecture notes by D. W.Anderson. W.A. Benjamin Inc., 1967.
[Bau]
T. Bauer.
Computation of the homotopy of tmf.
Available at
http://wwwmath.uni-muenster.de/u/tbauer/eo2ss.pdf.
[Bou96] A.K. Bousfield. On p-adic lambda rings and the k-theory of infinite
loop spaces. K-theory, 10:1-30, 1996.
[Bou99] A.K. Bousfield. The k-theory localizations and vl-periodic homotopy
groups of h-spaces. Topology, 38:1239-1264, 1999.
[CE99] Henri Cartan and Samuel Eilenberg. Homological Algebra. Princeton
University Press, 1999.
[Dwy38] P.S. Dwyer. Combined expansions of products of symmetric power
sums and of sums of symmetric power products with application to
sampling. The Annals of mathematical statistics, 9:1-47, 1938.
[GoeO4] M.J. Hopkins; P. Goerss. Moduli spaces of commutative ring spectra.
In Structured ring spectra, volume 315 of London Mathematical Society lecture notes series, pages 151-200. Cambridge University press,
2004.
[Gou88] F.Q. Gouvea. Arithmetic of p-adic modular forms. Number 1304 in
Lecture notes in mathematics. Springer Verlag, 1988.
[Hat01] A. Hatcher. Vector Bundles and K- Theory. online at authors website,
2001.
[Hir61] M.F. Atiyah; F. Hirzebruch. Vector bundles and homogenous spaces.
Proc. Sympos. Pure Math, 3:7-38, 1961.
[Kat75] N. Katz. L-functions via moduli of elliptic curves. Proc. Symp. Pure
math, 29:479-506, 1975.
[Kat76] N. Katz. p-adic interpolation of real analytic eisenstein series. Annals
of math, 104:459-571, 1976.
[Ken82] H.L. Keng. Introduction to Number Theory. Springer Verlag, 1982.
[Lau99] G. Laures. The topological q-expansion principle. Topology, 38:387425, 1999.
100
[Mac54] S. Eilenberg; S. MacLane. On the groups h(r,n)
computation. Annals of Math, 68:49-139, 1954.
2; methods of
[Mac79] I.G. Macdonald. Symmetric FUnctions and Hall Polynomials. Oxford University Press, 1979.
[Mil89] H.R. Miller. The elliptic character and the witten genus. In M. Mahowald and S. Priddy, editors, Algebraic topology, pages 281-289.
1989.
[Pri70]
S. Priddy. On w°Os°° and the infinite symmetric group. In Algebraic
Topology, volume 22 of Proc. Symp. Pure Math., 1970.
[Rez]
C. Rezk.
Supplementary
notes for math 512.
Available at
http://www.math.uiuc.edu/ rezk/512-spr2001-notes.pdf.
[Smi88] E.S. Devinatz; M.J.Hopkins; J.H. Smith. Nilpotence and stable ho-
motopy theory 1. Annals of math, 128:207-241,1988.
[Tal69] M.F. Atiyah; D.O. Tall. Group representations lambda rings and the
j-homomorphism. Topology, 8:253-297, 1969.
[Tor77] I. Madsen; V. Snaith; J. Tornehave. Infinite loop maps in geometric
topology. Mathematical proceedingsof the Cambridge philosophical
society, 81:399-430, 1977.
101