Internat. J. h. & M. Si.
Vol. 5 No. 4 (1982)675-690
675
INTRODUCTORY REMARKS ON COMPLEX MULTIPLICATION
HARVEY COHN
Department of Mathematics
City College of New York
New York N.Y. 10023 USA
(Received May 24, 1982)
ABSTRACT.
In its
Complex multiplication in its simplest form is a geometric tiling property.
advanced
form it is a unifying motivation of classical mathematics from el-
liptic integrals to number theory; and it is still of active interest.
This inter-
relation is explored in an introductory expository fashion with emphasis on a cen-
tral historical problem, the modular equation between
KEY WORDS AND PHRASES.
modular
functions
and
j(2z).
Complex multiplication, class number, elliptic integrals,
modular equation.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
i.
j(z)
10D05.
GEOMETRIC ILLUSTRATION.
We begin with an observation scarcely requiring ollege level mathematics:
"If a room has a rectangular floor plan in which the ratio of its sides
2, then
r
is
it may be partitioned into two rectangular rooms by a wall down the middle
of the longer side (see Fig. la) in such a fashion that each half has the same shape
as the original room (seen by using a 90
rotation)."
Clearly this is a unique phenomenon as stated.
still be present after division into two halves
(2/r
Only that ratio r
r),
and a
2
will
(nonrectangular)
parallelogram-shaped room could not permit this kind of a subdivision (since its
angles are not all equal).
Yet we can construct some kind of nontrivial parallelogram generalization, ann
once we
do
this there is no place to stop short of describing a major part of pure
mathematics’.
What is involved is a synthesis of number theory, algebra, analysis,
6 76
HARVEY COHN
ana topology which now all comes unaer the heaaing of complex multiplication.
Our
purpose is to offer a minimal escription of this phenomenon using only "generally
known" results.
2.
THE PARALLELOGRAM GENERALIZATION.
We replace the rectangle by a parallelogram with a perloalc structure (torus).
This means that the parallelogram may now be subaiviae into pieces which may be
move an reassemble accoralng to the group of vectors generate by the sies of
the parallelogram.
2X
(a)
(b)
0
(c)
FIGURE i
THREE COKPLEX
2
IK[LTIPLICATIONS OF NOP TWO
REMARKS ON COMPLEX MULTIPLICATION
For instance, in Fig. Ib, the square
A + B + C + D (on the right) can be re-
assembled as the rectangle
A* + B* + C + D, where
(or downward) translate of
A
(or
D + B*
cut in half to produce
B).
A*
and
677
(or
A*
This new rectangle
+ C, each of
B*)
is the right-hand
A *- + C + D + B*
can be
which has half the size but the
rotation).
same shape as the original (as seen by a 45
If this looks too obvious, consider the parallelogram on the right in Fig. ic.
It is
made up of isoscles
triangles
B + C
E + F
and
A + B + C + D + E + F
of sides
is equivalent to
(namely A* + B* + F
C*
and
i/8
rotation of arccos
A
triangles
+ E + D)
D
and
of sides
I, 4r2, 2.
A*
I, 2, 2
and
scalene
Thus the parallelogram
+ B* + C* + D + E + F
and
each half
is similar to the original (as seen by a
...).
69
When we introduce complex numbers we see that these are essentially the only
shapes for division of a parallelogram "modulo translations" into two equal similar
parts, and we also see why the configurations are described as "complex multlplications of norm
3.
two".
LATT ICE FORMULAT ION.
over
IR, namely Wl
and
H+
Using
Wl
and
Im
2/i
and
Wl
2
(T 6
> 0
is chosen to make
T
e
{nlwl + n22
[i’2
n l,n
(or the X-module of rank two generated by
The lattice
and
thought of either as
We write
(3.2)
7.]
and
2
in
).
-A
(3.3a)
[I,2]
A
where, symbolically, k
A
Wl
is
has many equivalent bases, i.e., it may be written as
L
e
matrices
2 6
lie in the upper-half plane
L, which
a set of points or as an abelian group of translations.
as
(3.1)
H+).
we generate the lattice
2
We define this ratio
(with a nonreal ratio).
w2
T
Thus the ordering of
(nonzero complex numbers) which are independent
K*
We consider two elements of
for a matrix
are identified).
kl alll + a122
k2 a21 I + a22w2
A
in
PSL2(.)
the modular roup,(the
In detail,
6 Z, A
(aij)
alla22 a12a21 1
aij
(3.3b)
678
HARVEY COHN
The determinant of
2/I E H+
A
is
+i (not +i) to preserve ordering, i.e., so that
The changes of bases are generated by the operations indicated
also
as follows:
(3.4)
PSL2(.)
The group
tails see Jacobson
is more complicated than
[12]
or Gunning
L,
and
(For further
nonabelian.
[9]).
The advantage of complex numbers comes with the concept of "shape" (or euclidean
It can be defined as the equivalence class of lattices
similarity).
[L}
{L I]
related by
L--LI,
which denotes the
iattice
Thus each
L
2]
[
where many different
[1,7]
I
T
[1,7],
Im
(e.g.,
,0
[i,0]
7
to
(3.6a)
> 0,
may be used, all related by
+
(all
0
(3.5b)
2
kl,
i/wl
is equivalent (using
L
Thus,
by
(3.3a),
[ i,
I
L
attained by multiplying each element of
in the symbols of (3.2) and
L
(3.5a)
E *,
PSL2(T.)(see
a12)/(a217 + a22)
(3.3)),
(3.6b)
We finally note the inclusion symbol
L
L0
for
a sublattice of
LI,
subgroup
areas of
t,
the index
L
0
c L
(3.7a)
i
is viewed as an additive
0
is easily interpreted as the (finite) ratio of
Actually, when
I
ILl/L01
L
parallelograms formed by the basis vectors of
L
I
and
L0o
Thus when
similarity occurs, we easily recognize
Ie/ El
II 2
(3.7b)
lhus we now verbalize Fig. 1 as the display of lattices
L
and
complex multi-
with norm given by
IL/ L!
We seem to be doing even better;
L
{I 2
2
(3.8)
we can arrange to have
[i’2 ]’
L
[1’2 2
(3.9)
For example, in Figs. labc (respectively we can recognize thi by using (3.4) as follows:
(3.10a)
REMARKS ON COMPLEX MULTIPLICATION
/-2[I,/-2/2]*
(i + i) [(l-i)/2, (l + i)/2]
[_2
(i + i)/2,
2
(i + i)[l,(l + i)/2]*
[2- ]
]
i,
I
(I + /-7)/2,
[-,I]*
[I,/-2]*
-2,-i]
I + i,
[2 ]
(3.10b)
[i + i,i]
i(
-’2
el
679
2
[l,i]
[l,l+i]*;
-2),
(3.i0c)
[- 2]*
(In each case, the analogues of (3.9) are innicate by
a*).
THE MULTIPLIER RING.
4.
R[L},
We now consider
R{L}
We include
RILl
0
p
IP L = L}
{p 6
OL
in the sense that
(4.1)
L.
is trivially in
=
L,
2L =
DlP2L = L
(Pl -+ P2 )L
L
pi e c L, p2 e c e
since
L
unner annition ann subtraction.
is closen
more, as the notation innicates,
(e ann
e,
R[L]
e
or the class
(4.2a)
e
c
(4.2b)
[e}.
If
Thus
R[e}
Z,
R[e}
(= m)
am
normalizen so that
a
a quadratic integer
is a quanratic
2
-bm + c
> 0, gcn(a,b,c)
,
n
b
2
t +
To sketch the proof, write
in
if and
2
(4.4)
4ac < 0,
i.e.,
important, with the same niscriminant
)
[ml,2]
L
Then the multiplier ring is generated by
z[]
(4.5)
satisfien a monic equation of integral trace
(
is callen the multi-
sur satisfying an equation over
0,
i.
R{e}
Here
Further-
it is seen to contain a
integers by innepennence of the basis vectors of L).
THEOREM 4.3. Complex multiplication occurs in a lattice
2/i
Z.
(Real multipliers must be
complex element, which performs complex multiplication.
only if the ratio
R[L}
From this,
really epenns on the equivalence class of
above, have the same multipliers).
plier rin of each
each
Thus it is clear that
is a ring:
pl L
[el,
L, i.e.,
the set of all multipliers of
n
[l,m]
t
and
norm n, and, more
n,
0,
n
t
[I,],
2
(4.6)
4n (< 0).
which has a subbasis in
[l,m]
for
R[L],
[l,m]
[bll + bl2m,b21
+
b22m]
c
[i,]
(4.7)
680
HARVEY COHN
where
bij
value"
and
monic for
6 Z, (but
.
[I,T]
But here
as an "eigen-
of the ring
RILl
The main difficulty is showing that
s.ame
discriminant
see the next section).
forms
We easily recognize
is not necessarily a generator
which necessarily leads to the
quadratic
(4.8)
as an "eigenvector", leading to equations of type (4.4) for
2, etc.).
may be
bll + bl2T
T b21 + b22T
bllb22 b12b21 I 12).
(it
done
by
Note the dependence
O[L} ,R[L]
d
RILl,
has a generator
(In practice this is
d.
T, and
R[O[L]].
2[]
(4.9)
We can show (4.9) by an explicit choice of generator
+ m, (m
i
Z) defined
by the cases
0 (mod
d _=
I (mod 4),
d _=
COROLLARY 4.11.
I 12
L
d/4
2
I
i
(4.10a)
0,
1)/4
(d
L
(4 10b)
0
occurs by
(6 R[L]
m[l,n0 2 ]’
2
n
o
/m
for which
is the maximum integer of
(4.12)
m n
R[L]
6
For proof, consider the elementary divisor form of the mapping
the factor
cases.
m
We can now
by itself is a real multiplication,
easy because they take place in lattices
d
-4
and
[i,]
-8, -4, -7
respectively.
limited number of
Note
for the interesting
1
The first few are
whose ring is just
Z[].
For norm
(4.13)
(3.10abc),
and
have discriminants
It can be verified that for a given norm n, only a
occur in (say)
(4.6), (since
t
2
4n +
d
< 4n).
CLAS S NUMBER.
We define
tices
Each
-L.
-3, namely
The complex multiplications of norm 2 are in
5.
m
L
(-i +/-3)/2
i,
d
and
proceed to enumerate all complex multiplications.
we have units for
with norm
can be chosen so that
[i’2
m
2
I
If complex multiplication of
n, then a basis of
where
4),
h
h(d), the class number of
LI,...,L h with complex
R{Li] is the same -[i
d
as the number of inequlvalent lat-
multiplication and the common dlscriminant d
the so-called principal lattice,
from (4.10ab), so we can always write L
and we
call
[L I]
I
d{Li].
[i,i]
the principal lattice class.
REMARKS ON COMPLEX MULTIPLICATION
To see a nonprincipal lattice class, we take
L
[I,-5], e 2
I
R{L I}
The fact that
ly"
L
and later on
R[L2]
-20, where
d
h
We have’
2.
[2,1 +-5],
(5.1a)
-5].
(5.1b)
is not in the principal class (of
2
681
L
I)
"algebraical-
is seen first
(see 6.9)) "transcendentally".
Historically lattices were treated by Gauss (1800) by a correspondance with hi-
nary positive quadratic forms
e
where
E I
t
an
+
d
b
(The details are omitted).
n
I
and
Borel etc.
n2,
[2]
This is done by writing
2
cn
+
a
> O, gcd(a,b,c)
I,
(5.2b)
4ac < 0
The forms lle in
h
PSL2(Z
equivalence classes under
which correspond uniquely to the classes
and
with
(5.2a)
.,
takes the form, in integers of
bnln 2
[w l,w2
L
tlnl01 + n2 212
(n l,n2)
is chosen so that
(nl,n 2)
on
(n l,n 2)
[L i].
(For etails, see
deSeguier [15]).
We bypass a wealth of technique to concentrate on the case (5.1ab) for
-20.
d
There
L
I
e2
n
+
5n22
l(nl’n2)
2n + 2nln + 3n22
(5.3a)
2(nl,n2).
(5.3b)
2
These forms must be inequivalent since they represent different sets of integers.
For instance,
2
does not
I
represents
22(ni n2)
i.e.
i
(and is indeed called prin_cipal accordingly, while
(2nl
+
n2)2 + 5n
2
for
(The term principal ideal arose eviously in this context; L I
.-5]
while
L
2
THEOREM 5.4.
is
I
and
n
.
in
2
is principal in
not).
For the discriminants
-28, -43, -67, -163
n
and
d
-3, -4, -7, -8, -ii, -12, -16, -19, -27,
more remarkably for no other, there is a unique equivalence
class of forms or lattices (h
I).
This result is the "Gauss-Heegner-Stark
Theorem", (see Stark [18]).
It is a
major illustration of the use of transcendental functions for algebraic purposes.
We shall
merely
define its man analytic tool, the modular invariant
j[L}
with
emphasis on the more manipulatively available aspects, rather than venturing into
topology
and
becoming absorbed in a different world, (e.g., of forms an Riemann
Surfaces, see Gunning [9]).
082
HARVEY COHN
THE MODULAR INVARIANT.
6.
j[L]
By definition, we are demanding that the modular invariant
in
T
H+),
(E
[I,T] E [e}.
if
j[L]
j(T)
a I’’i
+ a
I
+ a22J
T
2
a 21T
j
PSL (-), see
2
Thus we ask for invariance under
(3.6b),
a
., alla22-al
ij
be analytic
(6.1)
I
a
2 21
ImT >0.
We know from the generator properties in (3.4), that it suffices to show
J(T + 1)
j(-1/T)
(6.2)
j
To form this invariant, we start for symmetry’s sake with L
[l,m2 ].
We
consider the aggregate of lattice points
[nlw I + n2 21nl,n2
[]
L
define two invariants of
and
g2
g3
Z]
6
by double summation on
g2(ml’m2
g3(wl’m2
60
*
140 .*
denotes the omission from the sum of
constants
60
and
140
{e];
nevertheless,
3
2
and
n
2
(6.4a)
(6.4b)
(0,0).
(nl,n2)
are explained by (8.3) below).
g2/g3
I
-6
*
but not on
n
-4
where the
L
(6.3)
L
depend only on
Now
g2
[L]
and
(or on
(The strange
g3
depend on
ml/m2
With additional strange constants introduced we write
j
THEOREM 6.6.
j{e]
1728g
j(T)
/(g23
satisfies (6.1)
j(T)
The modular invariant
(6.5)
27g 3
and has the
further
properties that
(a)
3
g2
2
(b)
q
0),
27g3
poles) when
is analytic (free of
j(T)
and
j{L}
by the invariance
T
> 0, (in particular
is biunique onto
under
convergent for
exp 2 iT
Im
T --T
Im
T
+ i, J(T)
> 0,
and
is a Laurent series in
with integral coefficients starting
with
j(T)
any other analytic function of
(c)
bounds at
We
3
2
I/q + 744 + 196884q + 21493760q + 864299970q +
do
(<
constlql "m,
m 6
),
T
invariant under
(6.7)
PSL (.)
2
and with
polar
is a polynomial in
not attempt the proof here, although part
the connection with topology and Riemann Surfaces.
(c) is particularly the key to
We note the uncanny accuracy of
REMARKS ON COMPLEX MULTIPLICATION
(6.7) in computing
jr-2)
for the multipliers of
j
8000
20
3,
j(i)= 1728
12
3,
In addition in reference to (5.1ab), for
jr-5)= (50 + 26r5)
jl +2 /-5
3
=-153.
=-3375
(6.8)
=-20, we compute
(50- 26r5) 3
(6.9)
[e 2].
ILl]
which should further reassure us that
(6.9),
norm 2, namely,
j
d
683
It may seem precarious, in
to guess irrationals from decimals, but the sum and product of the entries in
(6.9) are integers.
THEOREM 6.10.
(4.10ab).
For every discriminant
Then the value of
I
d(< 0), let
denote
the generator in
is an algebraic integer of degree
j(l
h
over the
field
k
It generates a field over
J[Li}
is
(6.10b)
k0(J(l))
j(l
conjugates of
h
The
0.
l,...,h), the
(i
j(l
that
k
(6.10a)
2h), namely,
(of degree
K
which is abel ian over
(i ).
0rd)
0
k
over
various classes of discriminant
d.
real, it is a rational integer precisely when
Thus from the fact
i, (see (5.4)).
h
This is part of a startling result in number theory of Weber
(actually, "ring class field theory").
p
K
in
must, on reduction modulo
(1890)
We need the further definition that a prime
y
if for very element
are
0
of
K
the defining equation (over
linear factor implies all factors of the defining equation are linear since
normal.
We could even assume
has no multiple factors).
k
splits in
rd)
0
(excluding those
p
THEOREM 6.11.
to split in
form
K/
is
is restricted to integers whose defining equation
In the most familiar case (see Cohn [3])
d
is a quadratic residue of
d).
The result is as follows:
exactly when
which ivide
If the prime
K, see (6.10b),
Inl + n2 I 12
(One
p, contain only linear factors, possibly repeated.
p
is that we can represent
i.e., that we can solve for
n
and
I
p
p,(d/p)= I,
d, then the condition for
does not divide
p
we say that
p
by the principal quadratl.
n
2
in
Z
whichever
applies
Thus for
h(d
p
n
p
n
2
2
dn2/4
I
2
I
+
nln 2
d =_
(d
(6.12a)
0 (mod 4),
I)n22/4,
I, (6.12ab) follows exactly when p
d
-=
i (mod
(6.12b)
4).
satisfies (d/p)
I.
684
HARVEY COHN
[19] "almost"
Weber
field
k
by using
0
required to
traum").
j(), but a later modification of Fueter [6]
and
Hasse [I0] was
(Kronecker had called for such a construction as his "Jugend-
it.
do
constructed all fields relatively abelian over the base
Hilbert saw the phenomenon of a transcendental generator of algebraic
fields and formulated his famous "twelfth problem" (see Langlands
[14])
to investi-
Instead of describing the enormous complications to which this has
gate it further.
lead, we retreat to the complex multiplications of norm 2 to show how they affect
the j-function.
7.
THE MODULAR EQUATION.
n, j(n)
For any positive integer
equation with integral coefficients
and
we define three functions
and
Jl’ J2’
j(2),
jl()
can be seen to be a root of a polynomial
powers of
J3
j(/2),
j2()
Going directly to
2,
n
by
J3 ()
These are seen to be conjugates over the field of
J(
+2 )"
(7.1)
j(); indeed, we verify the fol-
2(Z) in (6.2)
Jl()’ Jl(-1/r) J2 ()
JB()’ J2(-1/) Jl ()
J2 ()’ J3(-1/r) J3 ().
PSL
lowing for the generators of
Jl ( +
J2 ( +
J3 ( +
j().
i)
1)
I)
(7.2a)
(7.2b)
(7.2c)
The last one is not quite immediate,
2T_
T
j(_2 +
(7.3)
From Theorem 6.6c we conclude that the symmetric functions of
roots of a polynomial equation with integral polynomials in
the modular equation of
order
de
y3
+
Y
X
and
(7.4)
[6],
Yui
[21])
24. 3. 31(X2y + y2X)_24.34.53(X2 + y2)
+34.53.4027XY + 28.37.56(X + y)-212.39.59.
On setting
as coefficients,
0
force will be the evaluation (see Fueter
X2y2
j()
are the
2,
F2(Ji, j())
Our final tour
3
F2(X,y X +
jl,J2,J3
(7.5)
factoring, we are reassured again by
-F2(X,X
(X-
203)(X + 153)2(X- 123).
(7.6)
REMARKS ON COIPLEX MULTIPLICATION
685
Here we have the three complex multiplications of norm 2 (see (6.8)), where
J(v)
j(2v), (see (3.9)).
F2(X,Y)
Before evaluating
reciprocal ole of
F2(J(2v)
because
J(4v)
Jl
j(2v)
j(v))
0
we note the symmetry in
and
and
J(v).
is a conjugate of
J2 j(v/2). The
F2(J(2v) j(4v))
Klein
[13]
X
Y, because of the
and
symmetry might worry us
0, leaving the impression that
To explain this "paradox" requires a careful accoun-
ting of the Galois group permutations of the
(seen in (7.2abc) to be 3-dihedral).
Ji
introduced his solutions of equations by polyhedra precisely in this
context, (the "icosahedron" entering for
8.
j(5), see Cohn [5]).
ELL IPT IC INTE GRAL S.
The classic way to forge a relation between
j()
and
j(2)
is to find two
elliptic integrals, one of which has periods in a lattice belonging to
L
Y
X
[l,W2 ])
(i.e., L’
and
the other of which has periods in a lattice belonging to
[wi,22]).
Oeffne (see Whittaker
of
We start with
Watson [20])
and
2
llu + *(ll(u
p(u)
z
[I,2
L
periods and constructing elliptic functions with these periods.
and
2,
Incredibly, the relations between such integrals are manag-
eable by "undergraduate calculus".
We use the Weierstrass method of starting with the lattice
(6.3)
(e.g.,
)2 ii2).
L
Then all elliptic functions with period lattice
are the field
(z,w), where,
(disdaining to discuss convergence), we set
Next, using the
2
4z
Thus, for any constant
and its
P(z)
from Theorem 6.6a).
g2 z
(8.3)
g3
dz/J4- 3
g2 z
du
( 0),
I
(8.4)
g3
is an elliptic integral of the first kind
3
g2 z
g3
[L},
as
has three distinct roots (since
Hence a typical period of
2
we find
periods form the lattices in the class
4z
e
2
z=e
3
dz/w
du
The polynomial
(6.4ab),
special coefficients in
w
(always finite)
* I/(u )3.
-2/u 3 -2
p’(u)
w
fdu
varies.
3
g2
2
2/g3
itself is
e
u
2e 2 =/e()
I
<8.5)
0
ILRVEY COHN
686
where, P(z)
is now a cubic (and later on a
(for a cubic,
blquadratic) with
also behaves like a root ).
The factor
e
e
and
I
2
as roots,
in (8.5) comes from the
2
fact that a period path does not really join two roots, but rather it encircles them
/P(z)
so that the value of
can return to its original sign.
Then the period path
contracts to a doubled value.
du
To manipulate
it is necessary to change it to the Legendre form so du
const dr, where
We can
by a linear transformation
do this
(z + 8)/(vz + 6), ,8,,6 6
z
Fortunately, this transformation
’k2Ze),
Ze)(l
dz/(l
dv
,
6
+ I.
k
o.
8
specified
need not be
(8.6a)
(8.6b)
All we need note is that we
are transforming the roots
(8.7a)
el, e2, e3,
of
4z3
g2 z
g3
into those of (I
r
(r
k
r
I
(e
x.
4.
r
4)/(r I
r
4)(r 3
r
2).
(8.7c)
k
e
2
4k/(k + 1)
2)
(8.7d)
associated with each other by permuting
These are also functions of
{k,llX,
1-
z,
surprise to obtain
j()
.
should return us to
g2
i,...,6,
j
and
g3"
(8.8)
Th.s it is no
as
.2
128
el, e2, e3,
k),(z-z)lk, x/(z- 1)},
11(1
so symmetric functions of the
j()
2)(r 3
e2)/(e 3
I
Actually, there are six values of
r
(8.b)
The cross-ratio of
to satisfy
k
rl, r2, r3,
namely,
is defined as
4
k
So we define
k2Z2),
2)(I
i, -i, l/k, -i/k
k for this by the cross-ratio.
We choose the right value of
rl, r2, r3,
Z
+ 384
2
256(k
16(k4 +
k +
14k
2
l)3/k2(k- 1) 2
+ l)3/k2(k 2 i) 4.
(8.9)
Now we use the Gauss-Landen transformation in (8.6a):
kZ-with
k’
chosen so
k
2
2/(Z’vrk’ + I/Z’rk’),
4k’/(k’ +
k
I)2,
(8. lOa)
or,
2/Qk’ + i/vrk’).
(8.10b)
REMARKS ON COMPLEX MULTIPLICATION
,
This transformation is
(k’ + I)
to
dv
dv
(remarkably)
one-to-two from
Z’,
to
(= dv/(k’ + I)).
A further property of the transformation is that the lattice
dv’(k’+l)
but
is the same as that of the lattice
double______.
in the other,
from
i
to
i/k
and
Finally, using (8.9)
Z’
k
in
j(2
L
for
I, Z
Z’
goes from
and
i (doubling the period).
Z’
1
z’
1//k’
Z
1/k
Z’
i/k’
Z
I.
k
14k
(8.13ab),
2
+
j(2T)
for
goes from
i
Thus in
to
L’.
-i to
i/k’, Z
goes
+
l)3/k2(k2
l)3/k ’2 (k ’2l)3/k4(k2 1) 2
1) 4
1) 4
(8.13a)
(8.13b)
we write
i/k)2/4;
(k2
64(s+4)3/s
64(4s+I) 3/ s
64(64/s
2
(8.14)
+ 48/s + 12 + s),
64(I/s + 12 + 48s + 64s
I/s
(8.15a)
2)
(’. 15b)
(only to be expected from the symmetry of
s
F2(X,Y) ).
j(2)
are now replaced by symmetric functions in
and
in
(8.12)
under
j()
i
(8.10ab),
There is a clear symmetry
So
wI
j(2), we have
2
16(k ’4 + 14k ’2 +
4
of periods of
Z= 1
(8.10b) for
16(k4 +
L’
becomes
On the other hand, when
back to
(8.11)
in the direction of
dv
to
s
j()
of
-i
256(k
To eliminate
L
goes from
j()
j(2)
j(T)
Thus
2
Specifically, we note that as
one-to-one fashion.
yet it transforms
where, after some hard work,
dZ’/(I-Z’2)(I k’2Z ’2)
dr’
:Z
687
t=s+ l/s,
(j() + j(2))/64
j(T)
j(2T)/642
64t
64t
The modular equation (7.5) (now in
3
2
+ 49t
(8.16a)
I16,
2
+ 816t + 3468t + 4913.
X + Y
and
XY)
(8.16b)
emerges from a hand-calculation
as the condition for a quadratic and a cubic polynomial (in
t) to have a common
root.
9.
SOME FOLKLORE AND HISTORY.
The discovery of complex multiplication is connected with a legend which
typically portrays C. F. Gauss as always the Ubermensch
and
never the Munchhausen
HARVEY COHN
688
At the age of 14, in 1791, Gauss in-
in his claims to prescience and omniscience.
the arithmetic-eometric mean (agm), as a mere curiousity.
vented
with two positive quantities
a’
n--
b’
b
and
transforms them by
and
(a + b)/2, b’ =rab
T(a’,b’), a’
Then
a
Here one starts
(a-b)2
has the order
(9.1)
T
so the iteration of
easily converges as
to
lira
Tn(a,b)
(M,M), M
agm(a,b).
(9.2)
1.198140234735592207439
(9.3)
Gauss calculated a few eases including
agm(l,r2)
Now, eight years later,
I
1799, Gauss [8] calculated
in
dx/vrl-x 4
and
naturally
0
took the ratio to its cognate
TT/2
dx
only to recall that this ratio
0
ii decimal places with the reciprocal of
agreed to
agm(l,2).
He justifie this
by proving that (under (9.1)),
I
/2 de/era cos2e+b2sin2 0/2 de/a’-2 cos2e+b,2 sinZe
2
(9.4)
0
/(2agm(a, b))
as it would follow by iterating
limit.
al
Thus for
Tn(a,b),
while
a
and
b
In terms of the integrals of type ($.6a), we can set
i
0
dZ/v/(i-Z 2)
a
lemniscate, i
i, b
2
+ r
(I-k22),
2, k
2cos
2
e
k
2
(a
2
b
approached a common
sin
Z
and find
2) /a 2
i, we obtain the arc length of the (figure-eight)
in polar coordinates.
by using the undoubled period in (.10ab).
Gauss prove (9.4), essentially,
The doubled period occurred even more
strongly in the form of theta-function identities (akin to the modular equation).
Remarkably, Gauss [7] did not publish this work until 1818, when he related it to
the problem in astronomy of approximating the perturbation of a slow planet oy a
fast planet, by replacing the fast planet’s orbit by an elliptical ring.
in
1795, J. Landen in Cambridge had essentially
tion of
(8.10ab), but presumably
Meanwhile,
produced the integral transforma-
not with the modular interpretation.
There is a conflicting view that the credit for complex multiplication must be
reserved to
N. H. Abel [i] who first
used
period lattices and complex multipliers,
REMARKS ON COMPLEX MULTIPLICATION
but only in 1828 (see Cohn
[4]).
complex multiplication by (i
689
Thus it would be within Abel’s scope to show
+ i)
by presenting the integral transformation of
Gauss and Landen in the form of the statement that the differential equation
dz/l
z
has the alKebraic integral in
z
4
z’ 4
-2i/(z’
2
I/z
(Some minor changes of variable are required).
,2)
(9.6b)
Euler had previously shown in I51
i
+ i),
credits. Fagnano who found such a formula as early as
1718:
how to produce a
(real) factor of
In this century, Hecke [II]
2
(9.6a)
z’
and
2
z
(i + i)dz’/l
(rather than
attempted to extend
but C. L. Siegel
[16]
Weber’s Theorem (6.11) to other
fields, but this led mostly to a theory of modular functions of several variables.
Weber’s original program is still being
G. Shimura
[17], R.
Langlands
[14],
and
pursued on a much more abstract level by
others, (but it can scarcely be given a
status report here).
ACKN(YLEDGMENTS.
Based on research supported by NSF Grant MCS 7903060.
versions of this worl. were presented at the University of Copenhagen in
1977
under
Earlier
January
a visit sponsored by the Danish National Research Council.
BIBLIOGRAPHICAL NOTE
An excellent exposition from an older vantage point was given by G. N. Watson
in the Mathematical Gazette
I(1933)5-I
agent, A Tale of the Eighteenth
under
Century".
the title "The Marquis
(He refers in the title
to
and
the Land-
Fagnano
and
Landen, respectively).
REFERENCES
i.
Abel, N.H. Recherches sur les fonctions elliptiques, Journ. reine angew. Math,
3(1828)160-190.
2.
Borel, A., Chowla, S., Herz, C.S., lwasawa,K, and Serre, J-P., .Seminar of
Complex Multiplication, Springer Lect. Notes., #21, 1966.
3.
Cohn, H., Second Course in Number Theory, Wiley, 1962, (Dover reprint, 1980,
under the title "Advanced Number Theory").
4.
Cohn, H., Diophantine equations over (t) and complex multiplication, (Number
Theory, Carbondale, 1979), Springer Lecture Notes #51, 1979, >0-81.
5.
Cohn, H., Iterated ring class fields
(1981) 107-122.
and
the icosahedron, Math.
Annalen., 225_
HARVEY COHN
090
6.
7.
8.
Fueter, R., Vorlesungen uber die Singularen Moduln und die Komplexe Multiplikation, I, II, Teubner, 1924, 192.
Gauss, C.F., Bestimmung der Anziehung eines elliptischen Ringes, Igl, (Reprint
by Ostwalds Klassiker, Leipzig, 1927).
Gauss, C.F., Gedenkband Anlassich des I00 Todestages am 23 Februar 1955,
Teubner, 1957.
Gunning, R.C., Lectures on Modular Forms, Princeton, 1962.
Hasse, H., Neue Begrundung
tier
komplexe Multiplikation, Journal reine a_ngew.
Math., 157(1927)115-139, 165(1931)64-,8.
ii.
Hecke, E., Hohere Modulfunktionen
und
ihre Anwendung auf die Zahlentheorie,
_Math. Annalen, __i(1912) 1-37.
Jacobson, N.., Basic Algebra_l
Klein, F.,
14.
er
Freeman, 1974.
die Transformation der elliptischen Funktionen und die
Aufl’-
sung funften Grades, _Math. Annalen, i__4(188)III-I>2.
Langlands, R., Some contemporary problems with origiixs: in the Jugendtraum,
(Mathematical Developments arising from Hilbert’s Problems), Proc. Sym_p:
Math.
15.
_P_u_r_e
Vol. 2, 1976, 401-41.
De Seguier, J., .S_u_r___deux Formules Fondamentales__ans _l_a _Thorie
des
Forms
Quadratiqu_e_s et de la Multiplication Complexe apres Kronecker, Berlin, 1894.
V.0_r_l_e_s._u_n_g.e._n_u_b_er__A_usewah_Ite
16.
Siegel, C.L.,
17.
Lecture Notes, Gottingen, 1965, (Engl. Trans. "Topics in Complex Function
Theory" Wiley 1971)
Shimura, G., A_utomorphic Functions and Number Theory, Springer Lecture Notes,
Kapitel der Funktionentheorie,
#54, 1968.
Stark, H.M., Class numbers of complex quadratic fields, (Modular Functions of
One Variable I, Antwerp, 19P2). pringer Lecture Notes, #320, 1973, 153-1>4.
Algebraische Zahlen., View, g, 1891.
19.
Weber, H., E_l__l_iPtische Funktionen
20.
Whittaker, E.T.,
21.
Yui, N., Explicit form of the modular equation, Jour. reine an&ew. Math.
and
und
Watson, G.N., _Course
299/300(197E) 185-200.
in Modern Analysis, Cambridge, 1940.