Selmer varieties
Minhyong Kim
20 Otober, 2008
Toronto
1
I. General bakground
(E; e) ellipti urve over Q .
G := Gal(Q =Q ).
The exat sequene
n
0!E [n℄!E (Q ) !
E (Q )!0
of groups with G-ation leads to the Kummer exat sequene:
n
1
0!E (Q )[n℄!E (Q ) !
E (Q ) !
H (G; E [n℄)
In fat, the boundary map indues an injetion
E (Q )=nE (Q ),!Hf1 (G; E [n℄);
where the subsript f refers to a subgroup of Galois ohomology
satisfying a olletion of loal onditions: A Selmer group.
2
Beause Hf1 (G; E [n℄) often admits an expliit desription, this
inlusion is applied to the problem of determining the group E (Q ).
Usually, we x a prime and run over its powers
E (Q )=pn E (Q ),!Hf1 (G; E [pn ℄)
leading to a onjetural isomorphism
E (Q ) Z p ' Hf1 (G; Tp (E ))
where
Tp (E ) := lim E [pn ℄
is the p-adi Tate module of E .
3
When X=Q is a urve of genus g 2 and b 2 X (Q ), analogue of
above onstrution
1
Z p ))
X (Q ) !
Hf (G; H1et (X;
uses the p-adi étale homology
Z p ) := 1et;p (X;
b)ab
H1et (X;
of X := X Spe(Q ) Spe(Q ).
Several dierent desriptions of this map.
4
But in any ase, it fators through the Jaobian
X (Q )!J (Q )!Hf1 (G; Tp J )
using the isomorphism
Z p ) ' Tp J;
H1et (X;
where the rst map is the Albanese map
x 7! [x℄ [b℄
and the seond is again provided Kummer theory on the abelian
variety J .
Consequently, diult to disentangle X (Q ) from J (Q ).
5
The theory of Selmer varieties renes this to a tower:
Hf1 (G; U4 )
-
?
Hf1 (G; U3 )
?
- Hf1(G; U2)
?
- 1
3 4
-
..
.
..
.
X (Q )
2
1
Hf (G; U1 )= Hf1 (G; Tp J Q p )
where the system fUn g is the Q p -unipotent étale fundamental group
b) of X .
1u;Q (X;
p
6
Brief remarks on the onstrutions.
1. The étale site of X denes a ategory
Q p)
Un(X;
of loally onstant unipotent Q p -sheaves on X . A sheaf V is
unipotent if it an be onstruted using suessive extensions by
the onstant sheaf [Q p ℄X .
2. We have a ber funtor
Q p )!VetQ
Fb : Un(X;
p
that assoiates to a sheaf V its stalk Vb . Then
U := Aut
(Fb );
the tensor-ompatible automorphisms of the funtor. U is a
pro-algebrai pro-unipotent group over Q p .
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3.
U = U1 U2 U3 is the desending entral series of U , and
Un = U n+1 nU
are the assoiated quotients. There is an identiation
Q p ) = V := Tp J Q p
U1 = H1et (X;
at the bottom level and exat sequenes
0!U n+1 nU n !Un !Un
1 !0
for eah n. For example, for n = 2,
^
0![ V=
2
Q p (1)℄!U2 !V !0:
8
4. H 1 (G; Un ) denotes ontinuous Galois ohomology with values in
the points of Un . For n 2, this is non-abelian ohomology, and
hene, does not have the struture of a group.
5. Hf1 (G; Un ) H 1 (G; Un ) denotes a subset dened by loal
`Selmer' onditions that require the lasses to be
(a) unramied outside a set T = S [ fpg, where S is the set of
primes of bad redution;
(b) and rystalline at p, a ondition oming from p-adi Hodge
theory.
9
6. The system
!Hf1(G; Un+1)!Hf1(G; Un)!Hf1(G; Un 1)! is a pro-algebrai variety, the Selmer variety of X . That is, eah
Hf1 (G; Un ) is an algebrai variety over Q p and the transition maps
are algebrai.
Hf1 (G; U ) = fHf1 (G; Un )g
is the moduli spae of prinipal bundles for U in the étale topology
of Spe(Z [1=S ℄) that are rystalline at p.
If Q T denotes the maximal extension of Q unramied outside T
and GT := Gal(Q T =Q ), then Hf1 (G; Un ) is naturally realized as a
losed subvariety of H 1 (GT ; Un ).
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For the latter, there are exat sequenes
Æ
0!H 1 (GT ; U n+1 nU n )!H 1 (GT ; Un )!H 1 (GT ; Un 1 ) !
H 2 (GT ; U n+1 nU n )
in the sense of ber bundles, and the algebrai strutures are built
up iteratively from the Q p -vetor spae struture on the
H i (GT ; U n+1nU n )
and the fat that the boundary maps Æ are algebrai. (It is
non-linear in general.)
So the underlying input from Arakelov theory is the niteness of
the ideal lass group, leading to nite-dimensionality of the
H i (GT ; U n+1 nU n ).
11
7. The map
na = fn g : X (Q ) - Hf1 (G; U )
is dened by assoiating to a point x the prinipal U -bundle
P (x) = 1u;Q (X ; b; x) := Isom
(Fb ; Fx )
p
of tensor-ompatible isomorphisms from Fb to Fx , that is, the
Q p -pro-unipotent étale paths from b to x.
For n = 1,
1 : X (Q )!Hf1 (G; U1 ) = Hf1 (G; Tp J Q p )
redues to the map from Kummer theory. But the map n for
n 2 does not fator through the Jaobian. Hene, the possibility
of separating the struture of X (Q ) from that of J (Q ).
12
8. If one restrits U to the étale site of Q p , there are loal analogues
1
na
p : X (Q p )!Hf (Gp ; Un )
that an be expliitly desribed using non-abelian p-adi Hodge
theory. More preisely, there is a ompatible family of isomorphisms
D : Hf1 (Gp ; Un ) ' UnDR =F 0
to homogeneous spaes for quotients of the De Rham fundamental
group
of X Q p .
U DR = 1DR (X Q p ; b)
U DR lassies unipotent vetor bundles with at onnetions on
X Q p , and U DR =F 0 lassies prinipal bundles for U DR with
ompatible Hodge ltrations and rystalline strutures.
13
Given a rystalline prinipal bundle P = Spe(P ) for U ,
D(P ) = Spe([P Br ℄G );
p
where Br is Fontaine's ring of p-adi periods. This is a prinipal
U DR bundle.
The two onstrutions t into a diagram
X (Q p )
- Hf1(Gp; U )
na
p
d na
r
=
r
- ?
DR 0
U
=F
whose ommutativity redues to the assertion that
1DR (X ; b; x) Br ' 1u;Q (X ; b; x) Br :
p
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9. The map
DR =F 0
na
:
X
(
Q
)
!
U
p
dr=r
Z is desribed using p-adi iterated integrals
1 2
n
of dierential forms on X , and has a highly transendental natural:
For any residue disk ℄y [ X (Q p ),
DR 0
na
dr=r;n (℄y [) Un =F
is Zariski dense for eah n and its oordinates an be desribed as
onvergent power series on the disk.
15
10. The loal and global onstrutions t into a family of
ommutative diagrams
- X (Q p )
X (Q )
?
?
-
- Hf1(Gp; Un) -D UnDR=F 0
lop
1
Hf (G; Un )
where the bottom horizontal maps are algebrai, while the vertial
maps are somehow transendental. Thus, the diult inlusion
X (Q ) X (Q p ) has been replaed by the algebrai map lop .
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Theorem 0.1
Suppose
dimHf1 (G; Un ) < dimHf1 (Gp ; Un )
for some n. Then X (Q ) is nite.
Remarks:
-Theorem is a rude appliation of the methodology. Eventually
would like rened desriptions of X (Q ) X (Q p ) as an amalgam of
the method of Chabauty and Coleman and the work of
Coates-Wiles, Kolyvagin, Rubin, Kato on the onjeture of Birh
and Swinnerton-Dyer.
-Hypothesis of the theorem expeted to always hold for n
suiently large. But diult to prove.
-Strategy is also inspired by an old onjeture of Lang.
17
Idea of proof: There is a non-zero algebrai funtion - X (Q p )
X (Q ) na
n
?
Hf1 (G; Un )
na
p;n
?
- Hf1(Gp; Un)
lo
p
96=0
?
Qp
vanishing on Im[Hf1 (G; Un )℄. Hene, Æ na
p;n vanishes on X (Q ).
But this funtion is a non-vanishing onvergent power series on
eah residue disk. 2
18
II. Polylogarithmi quotients and CM Jaobians.
The ompliated struture of U is an obstrution to diret usage.
However, there are quotients of U with simpler strutures. The
polylogarithm quotient of U is dened by
W := U=[U 2 ; U 2 ℄:
Also omes with a De Rham realization
W DR = U DR =[(U DR )2 ; (U DR )2 ℄;
and the previous disussion arries over verbatim.
But now, we an ontrol the dimension of Selmer varieties in a
larger number of ases.
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Suppose J is isogenous to a
produt of abelian varieties having potential omplex multipliation.
Choose the prime p to split in all the CM elds that our. Then
Theorem 0.2 (with John Coates)
dimHf1 (G; Wn ) < dimHf1 (Gp ; Wn )
for n suiently large.
Corollary 0.3 (Faltings' theorem, speial ase)
nite.
Applies, for example, to the twisted Fermat urves
axm + bym = z m
for a; b; 2 Q
n f0g and m 4.
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X (Q ) is
Preliminaries:
Need to ontrol
H 1 (GT ; W n+1 nW n )
as n grows. This leads via the exat sequenes
0!H 1 (GT ; W n+1 nW n )!H 1 (GT ; Wn )!H 1 (GT ; Wn 1 )
to ontrol of Hf1 (G; Wn ) H 1 (GT ; Wn ). That is,
X
dimH (G ; W
dimH (G; W ) 1
f
n
n
1
i=1
21
T
n+1 nW n ):
Sine W n+1 nW n is a usual Q p representation, we have the Euler
harateristi formula
dimH 0 (GT ; W n+1 nW n )
dimH 1 (GT ; W n+1 nW n )
+dimH 2 (GT ; W n+1 nW n ) = dim[W n+1 nW n ℄ :
But the H 0 term always, vanishes, so we get the formula
dimH 1 (GT ; W n+1 nW n ) =
dim[W n+1 nW n ℄ + dimH 2 (GT ; W n+1 nW n ):
22
A fairly simple ombinatorial analysis of the struture of W DR .
shows that
dimWnDR =F 0
X dim[W
n2g
(2g 2) (2g)! + O(n2g 1 ):
Meanwhile,
n
i=1
n2g
[(2g 1)=2℄ (2g)! + O(n2g 1)
i+1 nW i ℄
Sine g 2, we have
X dim[W
n
i=1
i+1 nW i ℄
X dimH (G ; W
<< dimWnDR =F 0 :
Therefore, it sues to show that
n
i=1
2
T
i+1 nW i ) = O(n2g 1 ):
23
Input from basi Iwasawa theory:
Let F=Q be a nite extension suh that all the CM is dened and
suh that F Q (J [p℄). We an enlarge T to inlude all the primes
that ramify in F . So we have
GF;T := Gal(Q T =F ) GT :
Beause the orestrition map is surjetive, it sues to bound
X dimH (G
n
i=1
2
F;T ; W
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i+1 nW i ):
If we examine the loalization sequene
X
0!
2 (W i+1 nW i )!H 2 (G
F;T
; W i+1 nW i )!
Y H (G ; W
2
vjT
v
i+1 nW i )
we see readily that
H 2 (Gv ; ) ' H 0 (Gv ; [W i+1nW i ℄ (1)) = 0
for i 6= 2. Thus, by Poitou-Tate duality, it sues to bound
X2(W i+1nW i) ' X1([W i+1nW i℄(1)):
The last group is dened by
X1([W i+1nW i℄(1))!H 1(GF;T ; [W i+1nW i℄(1))
0!
Y
! H (G ; [W
1
vjT
v
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i+1 nW i ℄ (1)):
By the Hohshild-Serre sequene, the group
X1([W i+1nW i℄(1))
is inluded in
Hom (M; [W i+1 nW i ℄ (1)) = Hom (M ( 1); [W i+1 nW i ℄ );
where
= Gal(F1 =F ) for the eld
F1 = F (J [p1 ℄)
generated by the p-power torsion of J and
M = Gal(H=F1 )
is the Galois group of the p-Hilbert lass eld H of F1 .
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Key fat (Greenberg following Iwasawa):
M is a nitely generated torsion module over the Iwasawa
algebra
:= Z p [[ ℄℄:
Let L 2 be an annihilator for M ( 1)=(Z p torsion). Thus, if we
knew an Iwasawa main onjeture for the Z rp -extension F1 =F , we
ould take L ould to be a redued multi-variable p-adi L-funtion.
For simpliity, we now assume that J itself has omplex
multipliation so that ' Z 2pg and
' Z p [[T1 ; : : : ; T2g ℄℄:
g be the haraters of G
Let fi g2i=1
F;T appearing in Tp J and
i = i .
27
The haraters that appear in [W i+1 nW i ℄ are all of the form
j1 j2 j3 ji ;
where j1 < j2 j3 ji ; eah with multipliity at most one.
For suh a harater to ontribute to Hom (M ( 1); [W i+1 nW i ℄ ),
we must have
j1 j2 j3 ji (L) = 0:
Furthermore, for eah suh harater, we have a bound
Hom (M ( 1); j1 j2 j3 j ) < B;
i
where B is the number of generators for M .
Thus the problem redues to ounting the number of zeros of L
among suh haraters.
28
The bulk of the ontribution omes from indies of the form
k < 2g j3 j4 ji :
So we an ount the zeros for the 2g
1 twists
Lk = L(k1(T1 + 1) 1; k2(T2 + 1) 1; : : : ; k;2g (T2g + 1) 1);
for kj = k (Tj + 1) 2g (Tj + 1), among
j3 ji
for dereasing sequenes (j3 ; : : : ; ji ) of numbers from f1; 2; : : : ; 2gg.
29
When we try to bound
X dimH (G ; W
n
i=2
2
T
i+1 nW i );
the possible multi-indies as i goes from 2 to n run over the lattie
points inside a simplex of edge length n 2 inside a 2g-dimensional
spae. Using a hange of variable one an always redue to L of the
form
L = a0(T1; : : : ; T2g 1) + a1(T1; : : : ; T2g 1)T2g + +al 1 (T1 ; : : : ; T2g 1 )T2lg 1 + T2lg :
From this formula, one easily dedues a bound O(n2g 1 ) for the
number of zeros.
2
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Remark:
Finiteness for ellipti urves follows the pattern
Non-vanishing of L-funtion ) niteness of Selmer group
) niteness of points.
For urves of higher genus with CM Jaobians, the impliations are
Sparseness of L-zeros
niteness of points.
) bounds for Selmer varieties )
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