Classi al Motives I: Motivi
L-fun
tions
Minhyong Kim
July 17, 2006
1
Variety V
; (V; s), -fun tion of V
; arithmeti
invariants of V .
This pi ture should be re ned by
Variety V
;
onstituent motives fMig
; f L(Mi; s)g, L-fun tions of the Mi
; arithmeti invariants of the Mi
; arithmeti invariants of V .
2
Example:
E=Q
tion
ellipti
urve with aÆne minimal equa-
y 2 + a1xy + a3y = x3 + a2x2 + a4x + a6
S:
set of primes of bad redu tion.
E : proper smooth model of E over Z [1=S ℄.
(E )0: set of losed points.
Y
1
S (E; s) =
1 N (x)
x2(E )0
s
Then there is a de omposition
S (E; s) = S (s)S (s
1)=LS (E; s)
3
where
S (s) =
and
Y
1
p2
=S
LS (E; s) =
Y
p2
=S
1
p s
Lp(E; s)
is the partial L-fun tion of E with fa tors
de ned by
1
Lp(E; s) =
1 app s + p1 2s
Here ap = p + 1 Np and Np is the number
of points on E mod p.
4
Can put in bad Euler fa tors a ording to a
re ipe determined by the redu tion of E at
p:
8
>
< 1=(1 p ss) split multipli ative;
Lp(s) = > 1=(1 + p ) non-split multipli ative;
:
1
additive:
L(E; s) :=
Y
p
Lp(E; s)
Using thus the breakdown into three fa tors,
we an also omplete S (E; s) in a natural
way.
5
The estimate japj 2pp implies that the
Euler produ t onverges for Re(s) > 3=2.
To ontrol the analyti properties, use relation to automorphi L-fun tions.
In this ase, an make expli it by omputing
the ondu tor
NE
Here
:=
Y
p2 S
p fp
fp = ordp(E ) + 1
mE
where E is the dis riminant of E and mE
is the number of omponents over F p of a
Neron model of E .
6
Fa t (W, T-W, BCDT): L has an analyti
ontinuation to the omplex plane.
In fa t,
L(E; s) = L(fE ; s)
Z1
1
s 1dy
=
f
(
iy
)
y
E
(2)s (s) 0
for a normalized weight 2 new usp form fE
of level NE whi h is an eigenve tor for the
He ke operators, determined by a q expansion
fp = 1 + a1q + a2q 2 + where the ap have to be the same as those
for E when p 2= S .
Can nd fE and then use this formula for
ompute L-values.
7
Conje ture (BSD):
ords=1L(E; s) = rankE (Q )
Proved if ords=1L(E; s) 1. (Kolyvagin)
8
Fun tional equation:
(E; s) := (2)s (s)NEs=2L(E; s)
satis es a fun tional equation
(E; 2 s) = E (E; s)
where E = 1 depends on the urve E .
Can be omputed in a straightforward way
as a produ t of lo al terms.
Now, if E = 1, then learly
L(E; 1) = 0
Suppose you an he k L0(E; 1) 6= 0 using
the equality with L0(f; 1), then we on lude
E (Q ) has rank one.
Thus, analysis of the L-fun tion, in luding
the fun tional equation and omputation,
gives us the stru ture of E (Q ).
9
Continuation of BSD: If
vanishing, then
(s 1)
= jSha(E )jRE
is the order of
r
js=1
r L(E; s)
Y
p
p=
jE (Q )(tor)j2
relating L-values to many other re ned arithmeti invariants of E .
General prin iple: L-fun tion en odes Diophantine invariants of E .
10
Brief dis ussion of terms.
Important distin tion: Rational terms versus trans endental terms.
Rational terms:
Sha(E ):
The Tate-Shafarevi h group of E ,
onje tured to be nite. Classi es lo ally
trivial torsors for E . Analogous to a lass
group.
E (Q )(tor ) :
p:
( nite) torsion subgroup of E (Q ).
Tamagawa number.
p
= (E (Q p ) : E 0(Q p ))
11
Trans endental terms:
RE :
Regulator of E omputed using anonial height <; > and basis fP1; P2; : : : ; Pr g for
E (Q )=E (Q )(tor).
:= j det(< Pi; Pj >)j
Thus, RE is the ovolume of the MordellWeil group, in a manner analogous to the
lassi al regulator of number elds ( ovolume of units).
RE
: real period
=
Z
j! j
where
!
and
= dx=(2y + a1x + a3)
<
>= H1(E (C ); Z )+
12
The known relations between L-fun tions
and arithmeti are expe ted to generalize
vastly.
L-fun
tions de ned using Galois a tions on
etale ohomology and ompleted using Hodge
theory.
13
Conje tures:.
(1) Hasse-Weil: analyti ontinuation and
fun tional equation, addressed by Langlands'
program: `Motivi L-fun tions are automorphi L-fun tions.'
(2) Values:
(a) Deligne generalizes dis ussion of period
(in non-vanishing ase) using omparison of
rational De Rham and topologi al ohomologies;
(b) Beilinson-Blo h generalizes dis ussion of
order of vanishing and regulator using rank
and ovolume of motivi ohomology.
( ) Blo h-Kato generalizes dis ussion of rational part using Tamagawa numbers for
Galois representations via p-adi Hodge theory.
14
X=Q :
smooth proje tive variety.
Asso iated to X is a olle tion of ohomology groups, the realizations of the motive
of X .
Q l ) for ea h prime l: the
= Hetn (X;
Q l - oeÆ ient etale ohomology of degree n.
Carries a natural a tion of = Gal(Q =Q ).
Hln(X )
:= H n(X; :X ): the algebrai De
Rham ohomology equipped with a Hodge
ltration given by
F iH n (X ) = H n(X; i),!H n (X )
n (X )
HDR
DR
for ea h i.
n (X )
HB
DR
:= H n(X (C ); Q ): the Q - oeÆ ient
singular ohomology of the omplex manifold X (C ) equipped with a ontinuous a tion F1 of omplex onjugation.
15
The ompleted L-fun tion of
all these stru tures.
H n(X )
uses
16
Canoni al omparison isomorphisms:
n (X )
HB
Ql
' Hln(X )
preserving a tion of F1.
n (X )
HB
C
n (X )
' HDR
C
This isomorphism endows HBn (X ) with a rational Hodge stru ture of weight n `de ned
over R .'
17
That is, we have a dire t sum de omposition
n (X ) C ' H p;q (X )
HB
where
H p;q := F p \ Fq
and
F1(H p;q ) = H q;p
If we denote by the omplex onjugation
on C then
(HBn (X )
C )F1
n
= HDR
R
18
At non-ar himedean pla es, there is an important analogue.
For any embedding Q ,!Q l , we have
DDR(Hln(X )) := (Hln(X )
n (X )
' HDR
BDR )
l
Ql
where l = Gal(Q l =Q l ), and BDR is Fontaine's
ring of p-adi periods.
19
Regardless of its pre ise de nition, a motive
M should have asso iated to it a olle tion
of obje ts as above that we all a pure system of realizations that make up a ategory
R.
That is, this is a olle tion
R(M ) = ffMlg; MDR; MB g
where ea h Ml is a representation of on
a ( nite-dimensional) Q l -ve tor spa e, MDR
is a ltered Q -ve tor spa e, and MB is a Q ve tor spa e with an involution F1. These
ve tor spa es should all have the same dimension and be equipped with a system of
omparison isomorphisms as above.
This data must be subje t to further onstraints having to do with lo al Galois representations.
20
Re all exa t sequen e:
v ^
0!Ip! p !
Z !0
where Ip is the inertia group and
^ ' Gal(F p=Fp):
Z
^ orresponds to the geometri FrobeF rp 2 Z
nius, that is, the inverse to the p-power
map.
For l 6= p, Ip has a tame l-quotient
tl : Ip!Ip;l
with the stru ture
^l (1) ' lim ln
Ip;l ' Z
as a module for Gal(F p=Fp).
De ne
Wp := v 1(Z )
the Weil group at p.
p;
21
Convenient to analyze the data of Ml using
an asso iated Weil-Deligne (W-D) representation
W Dp(Ml )
for ea h p, onsisting of
-a representation r of Wp su h that rjIp has
nite image,
-and a nilpotent operator
representation.
Np
a ting on the
These satisfy a ompatiblity
r(p)Npr(p 1) = p 1Np
for any lift p 2 Wp of F rp.
22
The onstru tion of W Dp(Ml) for p 6= l uses
the fa t that the a tion of Ip when restri ted
to some nite index subgroup Ip0 is unipotent, and hen e, an be expressed as
7! exp(tl()Np)
for a nilpotent Np. Then the representation
r is given by
r(np ) = np exp( tl ( )N )
For p = l, we use the fa t that any De Rham
representation is potentially semistable, and
hen e, gives us a ltered (l; Nl) module via
Ml
7! (Ml
Bst)
0
l
whi h is, in any ase, isomorphi to MDR.
23
Remarks:
-The point of this onstru tion is that we
an pa kage the information of the representation in a form that does not use the
topology of Q l . Thereby makes natural the
onne tion to omplex automorphi forms.
-Creates a pre ise analogy with limit mixed
Hodge stru tures.
-We an de ne the
Frobenius semi-simpli ation W Dp(Ml)ss
of W Dp(Ml) by repla ing
simple part.
p
with its semi-
24
Here are the onstraints we impose on our
pure system of realizations:
-We assume then that there exists a nite
set S of primes su h that W Dp(Ml) is unrami ed for all p 2= S , i.e., Np = 0 and Ip
a ts trivially.
-`Algebrai ity and independen e of l':
There exists a Frobenius semi-simple W-D
representation W Dp(M ) over Q su h that
W Dp(M ) Q l ' W Dpss(Ml ) Q l
for any embedding
Q ,!Q l
Subje t to these onditions, the olle tion
fMlg is then referred to as a strongly ompatible system of l-adi representations.
25
-`Weil onje ture':
There should exist an integer n, alled the
weight of M , su h that the eigenvavlues
of F rp a ting on W Dp(M ) for p 2= S have
all Ar himedean absolute values equal to
pn=2. Furthermore, the Hodge stru ture MB
should be pure of weight n.
-`purity of monodromy ltration': If we denote by Mn: the unique in reasing ltration
on W Dp(M ) su h that Mn k = 0, Mnk =
W Dp(M ) for suÆ iently large k and
N (Mnk ) Mnk 2;
then the asso iated graded pie e
GrkMn(W Dp(M ))
has all Frobenius eigenvalues of ar himedean
absolute value p(n+k)=2.
26
Remarks:
-In general, need to allow oeÆ ients in E
for the representations where E is a number
eld and E are ompletions. Arise naturally
when onsidering dire t summands or motives with oeÆ ients, e.g., abelian varieties
with CM.
-The bi-grading
MB
C
' M p;q
whi h is ompatible with the omplex onjugation of oeÆ ients orresponds to a representation of the group
ResCR (Gm )
27
-Together with the a tion of
F1 Æ C
it an be viewed as a representation of the
real Weil group with points given by
WR (R ) = C [ C j
where j 2 = 1 and jzj 1 = z.
Here, C is the Weil operator de ned by
C jM pq
= iq
p
-It is onje tured that the realizations
n (X ); H n (X ))
H n(X ) = (fHln(X )g; HB
DR
oming from a smooth proje tive variety X
satisfy the algebrai ity, independen e of l,
and purity onditions even for p 2 S .
28
Category of pure motives should be omprised of obje ts in R of geometri origin, a
notion with a rather pre ise interpretation.
For example, need to allow duals (homology) and tensor produ ts of all obje ts onsidered.
Obje ts that are not generated in an obvious way from those of the form
H n(X )
arise via images (or kernels) under pull-ba ks
and push-forwards in ohomology indu ed
by maps of varieties, as well as Q -linear ombinations of geometri maps.
Also should be able to ompose pull-ba ks
with pushforwards.
29
Su h ompositions give rise to the idea of
using orresponden es modulo homologi al
equivalen e as morphisms.
On e morphisms are onstru ted in this manner, get naturally new obje ts using the deomposition of
End(H n(X ));
whi h is a semi-simple Q -algebra subje t to
one of the standard onje tures that numeri al equivalen e and homologi al equivalen e oin ide.
30
Can onsider a ategory of mixed systems of
realizations by requiring a weight ltration
Wn
1M
WnM Wn+1M ompatible with all the omparisons and su h
that ea h graded quotient
n (M )
GrW
is a pure system of realizations of weight n.
Mixed motives should be those of geometri origin su h as the ohomology of varieties that are not ne essarily smooth or
proper. But then, need to in lude obje ts
like ( nite-dimensional quotients of)
Q [1 ℄
or the ( o)-homology of ( o-)simpli ial varieties.
31
Given a pure system M of realizations we
an de ne its L-fun tion L(M; s) as an Euler
produ t
L(M; s) =
with
Y
p
Lp(M; s)
Lp(M; s)
=
det[(1
1
p sF rp)j(W Dp(M ))Ip=1;Np=0℄
Assume M is of weight n, then produ t onverges (and hen e is non-zero) for
Re(s) > n=2 + 1:
32
Also a fa tor at 1 depending upon the representation MB C of WR .
De ne
s=2
R
:= C
:= 2(2)
(s=2)
s
(s)
:= dimM pq
hp; := dimM pp;1
hpq
where the signs in the supers ript refer to
the 1 eigenspa es of the F1-a tion.
33
Then
L1(M; s)
is de ned by
Y
p<q
C (s
p)h
for odd n, and
Y
p<q
h
C (s p )
pq
R (s
n=2)h
n=
pq
2+
R (s
n=2+1)h
for n even.
34
2
n=
It is onje tured that (M; s) has a meromorphi ontinuation to C and satis es a
fun tional equation
(M; s) = (M; s)(M ; 1 s)
where the epsilon fa tor has the form (M; s) =
bas. This onje ture should be addressed by
the Langlands' program.
35
Notation:
Q:
trivial system of realizations.
Q (1)
:= H 2(P1)
Q (i)
= Q (1)
i
for i 0
and Q (i) = Hom(Q ( i); Q ) for i < 0.
For a system M of realizations,
M (i) := M Q (i)
Then for any smooth proje tive variety of
dim d, we have
H 2d(X ) ' Q ( d)
and a perfe t pairing
H i(X ) H 2d i(X )!H 2d(X )
Cup produ t with the ohomology lass of
a hyperplane gives us
H i(X ) ' H 2d i(X )(d i)
36
Some simple properties of twisting:
M (n)l is the tensor produ t of Ml with the
n-th power of the Q l y lotomi hara ter.
F i(M (n)DR) = F n+iMDR
with a orresponding shift in Hodge numbers hpq .
F1jM (n)B
= (F1jMB ) ( 1)n
Finally,
L(M (n); s) = L(M; s + n)
37
Hen eforward, we will fo us on the ase
where M is H n(X ) for a smooth proje tive
variety X of dimension d and assume that
- H n(X ) is a pure system of realizations;
- the analyti ontinuation and fun tional
equation hold true.
38
Conje tures on orders.
We have
H n(X ) ' H 2d n(X )(d) ' H n(X )(n)
Thus, the fun tional equation relates
L(H n(X ); s)
and
L(H n(X )(n); 1
s) = L(H n(X ); n + 1
s)
with enter of re e tion
(n + 1)=2
Thus, for the most part, we an on ne
interest to
m (n + 1)=2
or, equivalently,
n+1
m (n + 1)=2:
39
Brief reminder on two simple ase:
n
odd: BSD
ords=1L(H 1(E ); s) = rankE (Q )
Now, an element
x 2 E (Q )
gives rise to an extension in the ategory R
of realizations
Æ (x) 2 Ext1R(Q ; H 1(E )(1))
It is onje tured that when R is repla ed by
a suitable ategory of motives, this is the
only way to onstru t su h extensions.
40
n
even:
F=Q
Galois extension and
: Gal(F=Q )!Aut(V )
a nite-dimensional representation.
L(; s)
Artin L-fun tion. Then
ords=1L(; s) =
dimHomRep(Q ; V )
41
The general onje ture is
ords=n+1 mL(H n(X ); s)
= dimExt1MotZ (Q ; H n(X )(m))
dimHomMotZ (Q ; H n(X )(m))
The Hom and Ext should o ur inside a onje tural ategory of mixed motives over Z
with Q - oeÆ ients.
For weight reasons, the Hom term vanishes
unless n = 2m in whi h ase the Ext term
vanishes. That is, in the pure situation we
are onsidering, only one term or the other
o urs.
This is the prototype of the sort of statement that should hold for an arbitrary (mixed)
motive.
42
So when n = 2m, this be omes
ords=m+1L(H 2m (X ); s) =
dimHomMotZ (Q ; H 2m(X )(m))
generalizing he pole of the Artin L-fun tion
(m = 0).
It is expe ted that
HomMotZ (Q ; H 2m(X )(m))
' [CH m(X )=CH m(X )0℄
Q
Of ourse the isomorphism should arise via
a y le map
CH m(X )!H 2m(X )(m)
killing the y les CH m(X )0 homologi ally
equivalent to zero.
43
When n + 1 = 2m, the onje ture predi ts
the order of vanishing at the entral riti al
point:
ords=mL(H 2m 1(X ); s)
= dimExt1MotZ (Q ; H 2m 1(X )(m))
It is then onje tured that
dimExt1MotZ (Q ; H 2m 1(X )(m)) ' CH m(X )0
44
Q
The map from y les to extensions goes as
follows: given a representative Z for a lass
in CH m(X )0, we get an exa t sequen e
0!H 2m 1(X )(m)!H 2m 1(X n Z )(m)
!Æ HZ2m(X )(m)!H 2m(X )(m)
There is a lo al y le lass
l(Z ) 2 HZ2m(X )(m)
that maps to zero in H 2m(X )(m), giving rise
to the desired extension:
0!H 2m 1(X )(m)!Æ 1( l(Z ))!Q !0
45
These two lassi al points, entral riti al:
n+1
m = m = (n + 1)=2;
n
odd;
and just right of it:
n+1
m = n=2 + 1;
even;
are somewhat ex eptional. In all other ases,
one expe ts
n
n+1
Ext1MotZ (Q ; H n(X )(m)) = HM;
Z (X; Q (m))
with the last group, often referred to as motivi ohomology, de ned using K -theory :
Im[(K2m n 1(X ))(m) !(K2m n 1(X ))(m) ℄
(X is a proper at regular Z -model for
or Blo h's higher Chow groups
Im[CH n+1(X ; 2m
!CH n+1(X; 2m
n
1)
X)
Q
1) Q ℄
Latter interpretation more popular lately.
n
46
However, intrinsi interpretation in terms of
the ategory of motives should be kept in
mind in all onstru tions.
In fa t, when m > n=2 + 1, the onje tured
fun tional equation implies
ords=n+1 mL(H n(X ); s)
= dimExt1MHS R (R ; HBn (X )(m)
R)
R
where the extension o urs inside the ategory of real mixed Hodge stru tures de ned
over R . So the onje ture on order of vanishing follows from the onje ture that the
Hodge realization fun tor indu es an isomorphism
Ext1MotZ (Q ; H n(X )(m))
' Ext1MHSR (R ; HBn (X )(m)
R
R
R)
47
In general, onje ture should be on eptualized in two parts:
(1) Relation between L fun tions and
groups in ategory of motives.
Ext
(2) Geometri interpretation of Ext groups.
Provides unity to a wide range of related
issues in Diophantine geometry.
48
There is a onstru tion, onvenient in pra ti e, of the real Ext group via Deligne ohomology:
n (X )(m)
Ext1MHS R (R ; HB
R
R)
' HDn+1(XR ; R (m))
and using properties of Deligne ohomology,
one an onstru t regulator maps
n+1
HM;
Z (X; Q (m))
!Ext1MHSR (R ; HBn (X )(m)
R
R)
that an be studied independently of a ategory of motives.
For example, an onstru t subgroups
n+1
L HM;
Z (X; Q (m));
that should onje turally be of full rank, and
study their image.
49
Conje tures on trans endental part of values.
Central riti al values (Blo h-Beilinson): s =
m, n = 2m 1.
We have an isomorphism
2m 1(X )
F m HDR
' [HB2m 1(X )(m 1)℄(
R
1)m 1
R
This is then realized as an isomorphism
2m 1(X ))℄ 1
[^top(F mHDR
^
top[[H 2m
B
1(X )(m
m 1
(
1)
℄
1)℄
R
'R
50
Choosing bases for the two Q -lines determines a period
p(H 2m 1 (X )(m)) 2 R =Q Get additional trans endental ontribution
by onsidering a height pairing, onje tured
to be non-degenerate:
CH m(X )0 CH dim(X )+1 m(X )0!R
whose determinant gives us a regulator
r(H 2m 1(X )(m)) 2 R =Q Re all that onje turally
dm
:= ords=mL(H 2m 1(X ); s)
= dimCH m(X )0
Q
51
As for the value then, it is onje tured that
L(H 2m 1(X ); m)
:= slim
!m(s
m) dm L(H 2m 1(X ); s)
= p(H 2m 1(X )(m))r(H 2m 1 (X )(m))
in R =Q .
52
Values at n + 1
m < n=2.
Note that this is equivalent to
m > n=2 + 1;
the region of onvergen e for the Euler produ t. Hen e, we are skipping the lassi ally
interesting ase of
m = n=2 + 1
(n + 1
m = n=2)
for n even.
Instead of a period isomorphism, there is
then an exa t sequen e:
m 1
(
1)
n
m
n
R℄
0!F H (X ) R ![H (X )(m 1)
B
DR
!Ext1MHSR (R ; HBn (X )(m)
R
R)
!0
53
Thus, the trans endental part should in orporate a Q -stru ture on
n (X )(m)
Ext1MHS R (R ; HB
R
R)
oming from the onje tured isomorphism
n+1
HM;
Z (X; Q (m)))
R
' Ext1MHSR (R ; HBn (X )(m)
R
R)
Assuming this, we are led to a trivialization
n (X )℄ 1 ^top([H n (X )(m 1)℄( 1)m 1 )
[^topF mHDR
B
n+1
1
(
X;
Q
(
m
)))℄
[^top(HM;
Z
R
'R
54
Thus, hoosing bases for the three Q -lines
determines a number
(H n(X )(m)) 2 R =Q :
Beilinson's onje ture is that
L(H n(X ); n + 1 m)
= (H n(X )(m)):
55
For the value at m = n=2 (n + 1 m =
n=2 + 1), the regulator in orporates maps
both from motivi ohomology
and
n+1
HM;
Z (X; Q (m + 1))
CH m(X )0:
56
In the Blo h-Kato onje ture isomorphisms
are normalized more arefully, omparing ertain integral stru tures one prime at a time.
Thereby, it onstru ts a lift of (H n(X )(m))
to R and interprets
q (H n(X )(m)) :=
L(H n(X ); n + 1
m)= (H n(X )(m))
in terms of arithmeti invariants arising from
Galois ohomology.
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Extra tion of the rational part is supposed
to lead eventually to a p-adi L-fun tion
L(p)(H n(X ))
that exer ises ontrol over Galois ohomology (i.e., Selmer groups) and Diophantine
invariants.
Best strategy so far for `dire t appli ation'
of L-fun tions to the elu idation of Diophantine stru tures.
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Warning: onspi uous de ien y in theory
of motives:
Even in the best of possible worlds, only
abelian invariants are a essible, su h as
CH m(X ):
Does not yield information about
X (Q )
unless X is an abelian variety.
In fa t, theory of motives is impli itly modelled after the theory of abelian varieties and
H1.
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Attempts to address this de ien y for ertain varieties are ontained in
Grothendie k's anabelian program
that on erns itself with the theory of
pro- nite 1's:
Also an interesting role for the intermediate theory of motivi fundamental groups,
where Ext groups are repla ed by lassifying
spa es for non-abelian torsors.
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