Selmer varieties
Minhyong Kim
18 November, 2008
Cambridge
1
I. General ba kground
(E; e) ellipti urve over Q .
G := Gal(Q =Q ).
The exa t sequen e
n
0!E [n℄!E (Q ) !
E (Q )!0
of groups with G-a tion leads to the Kummer exa t sequen e:
n
1
0!E (Q )[n℄!E (Q ) !
E (Q ) !
H (G; E [n℄)
In fa t, the boundary map indu es an inje tion
E (Q )=nE (Q ),!Hf1 (G; E [n℄);
where the subs ript f refers to a subgroup of Galois ohomology
satisfying a olle tion of lo al onditions: A Selmer group.
2
Be ause Hf1 (G; E [n℄) often admits an expli it des ription, this
in lusion 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 onje tural 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 onstru tion
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 des riptions of this map.
4
But in any ase, it fa tors through the Ja obian
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 se ond is again provided Kummer theory on the abelian
variety J .
Consequently, di ult to disentangle X (Q ) from J (Q ).
Eorts of Weil, Mumford, Vojta.
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 onstru tions.
1. The étale site of X denes a ategory
Q p)
Un(X;
of lo ally onstant unipotent Q p -sheaves on X . A sheaf V is
unipotent if it an be onstru ted using su essive extensions by
the onstant sheaf [Q p ℄X .
2. We have a ber fun tor
Q p )!Ve tQ
Fb : Un(X;
p
that asso iates to a sheaf V its stalk Vb . Then
U := Aut (Fb );
the tensor- ompatible automorphisms of the fun tor. U is a
pro-algebrai pro-unipotent group over Q p .
7
3.
U = U1 U2 U3 is the des ending entral series of U , and
Un = U n+1 nU
are the asso iated quotients. There is an identi ation
Q p ) = V := Tp J
U1 = H1et (X;
at the bottom level and exa t sequen es
0!U n+1 nU n !Un !Un
1 !0
for ea h n. For example, for n = 2,
^
0![ V=
2
Q p (1)℄!U2 !V !0:
8
Qp
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
hen e, does not have the stru ture of a group.
5. Hf1 (G; Un ) H 1 (G; Un ) denotes a subset dened by lo al
`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 redu tion;
(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, ea h
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 spa e of prin ipal 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 ).
10
For the latter, there are exa t sequen es
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 stru tures are built
up iteratively from the Q p -ve tor spa e stru ture on the
H i (GT ; U n+1nU n )
and the fa t that the boundary maps Æ are algebrai . (It is
non-linear in general.)
So the underlying ar himedean input 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 asso iating to a point x the prin ipal 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)
redu es to the map from Kummer theory. But the map n for
n 2 does not fa tor through the Ja obian. Hen e, the possibility
of separating the stru ture of X (Q ) from that of J (Q ).
12
8. If one restri ts U to the étale site of Q p , there are lo al analogues
1
na
p : X (Q p )!Hf (Gp ; Un )
that an be expli itly des ribed using non-abelian p-adi Hodge
theory. More pre isely, there is a ompatible family of isomorphisms
D : Hf1 (Gp ; Un ) ' UnDR =F 0
to homogeneous spa es for quotients of the De Rham fundamental
group
U DR = 1DR (X
of X
Q p ; b)
Q p.
U DR lassies unipotent ve tor bundles with at onne tions on
X Q p , and U DR =F 0 lassies prin ipal bundles for U DR with
ompatible Hodge ltrations and rystalline stru tures.
13
Given a rystalline prin ipal bundle P = Spe (P ) for U ,
D(P ) = Spe ([P
B r ℄G );
p
where B r is Fontaine's ring of p-adi periods. This is a prin ipal
U DR bundle.
The two onstru tions t into a diagram
X (Q p )
- Hf1(Gp; U )
na
p
d na
r
=
r
- ?
DR 0
U
=F
whose ommutativity redu es to the assertion that
1DR (X ; b; x) B r ' 1u;Q (X ; b; x) B r :
p
14
9. The map
DR =F 0
na
:
X
(
Q
)
!
U
p
dr= r
Z
is des ribed using p-adi iterated integrals
1 2 n
of dierential forms on X , and has a highly trans endental natural:
For any residue disk ℄y [ X (Q p ),
DR 0
na
dr= r;n (℄y [) Un =F
is Zariski dense for ea h n and its oordinates an be des ribed as
onvergent power series on the disk.
15
10. The lo al and global onstru tions t into a family of
ommutative diagrams
- X (Q p )
X (Q )
?
?
-
- Hf1(Gp; Un) -D UnDR=F 0
lo
1
Hf (G; Un )
p
where the bottom horizontal maps are algebrai , while the verti al
maps are somehow trans endental. Thus, the di ult in lusion
X (Q ) X (Q p ) has been repla ed by the algebrai map lo p .
16
Theorem 0.1
Suppose
D Æ lo p (Hf1 (G; Un )) UnDR =F 0
is not Zariski dense for some n. Then X (Q ) is nite.
Remarks:
-Theorem is a rude appli ation of the methodology. Eventually
would like rened des riptions of the image of the global Selmer
variety, and hen e, of X (Q ) X (Q p ) by extending the method of
Chabauty and Coleman and the work of Coates-Wiles, Kolyvagin,
Rubin, Kato on the onje ture of Bir h and Swinnerton-Dyer.
-Strategy is also inspired by an old onje ture of Lang.
17
Idea of proof: There is a non-zero algebrai fun tion
- X (Q p )
X (Q ) na
n
?
Hf1 (G; Un )
na
p;n
?
- UnDR=F 0
DÆlo
p
9 6=0
?
Qp
vanishing on Im[Hf1 (G; Un )℄. Hen e, Æ na
p;n vanishes on X (Q ).
But this fun tion is a non-vanishing onvergent power series on
ea h residue disk. 2
18
-Hypothesis of the theorem expe ted to always hold for n
su iently large, but di ult to prove. For example, Blo h-Kato
onje ture on surje tivity of p-adi Chern lass map, or
Fontaine-Mazur onje ture on representations of geometri origin
all imply the hypothesis for n >> 0.
That is, Grothendie k expe ted
Non-abelian `niteness of
niteness of X (Q ).
X' (= se tion
Instead we have:
`Higher abelian niteness of
onje ture) )
X' ) niteness of X (Q ).
19
Suppose J is isogenous to a
produ t of abelian varieties having potential omplex multipli ation.
Choose the prime p to split in all the CM elds that o ur. Then
Theorem 0.2 (with John Coates)
D Æ lo p (Hf1 (G; Un )) UnDR =F 0
is not Zariski dense for n su iently large.
20
Corollary 0.3 (Faltings' theorem, spe ial
hypothesis of the theorem, X (Q ) is nite.
ase)
Applies, for example, to the twisted Fermat urves
axm + bym = z m
for a; b;
2 Q n f0g and m 4.
21
With the
Idea: Constru t a quotient
U !W !0
and a diagram
- X (Q p )
X (Q ) na
n
na
p;n
?
lo
?
lo
Hf1 (G; Un )
Hf1 (G; Wn )
?
- Hf1(Gp; Un) D- UnDR=F 0
p
?
- Hf1(Gp; Wn)
p
22
?
- WnDR=F 0
D
su h that
dimHf1 (G; Wn ) < dimWnDR =F 0
for n >> 0.
23
II. Polylogarithmi
quotients and CM Ja obians.
The ompli ated stru ture of U is an obstru tion to ontrolling
Selmer varieties. However, there are quotients of U with simpler
stru tures. 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 dis ussion arries over verbatim.
But now, we an ontrol the dimension of Selmer varieties in a
larger number of ases.
24
Suppose J is isogenous to a
produ t of abelian varieties having potential omplex multipli ation.
Choose the prime p to split in all the CM elds that o ur. Then
Theorem 0.4 (with John Coates)
dimHf1 (G; Wn ) < dimWnDR =F 0
for n su iently large.
25
Preliminaries:
Need to ontrol
H 1 (GT ; W n+1 nW n )
as n grows. This leads via the exa t sequen es
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
26
T
n+1 nW n ):
Sin e W n+1 nW n is a usual Q p representation, we have the Euler
hara teristi 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 ):
27
A fairly simple ombinatorial analysis of the stru ture 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 ℄
Sin e g 2, we have
X dim[W
n
i=1
i+1 nW i ℄
X dimH (G ; W
<< dimWnDR =F 0 :
Therefore, it su es to show that
n
i=1
2
T
i+1 nW i ) = O(n2g 1 ):
28
Input from basi Iwasawa theory:
Let F=Q be a nite extension su h that all the CM is dened and
su h that F Q (J [p℄). We an enlarge T to in lude all the primes
that ramify in F . So we have
GF;T := Gal(Q T =F ) GT :
Be ause the orestri tion map is surje tive, it su es to bound
X dimH (G
n
i=1
2
F;T ; W
29
i+1 nW i ):
If we examine the lo alization sequen e
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 su es 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
30
i+1 nW i ℄ (1)):
By the Ho hs hild-Serre sequen e, the group
X1([W i+1nW i℄(1))
is in luded 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 .
31
Key fa t (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 onje ture for the Z rp -extension F1 =F , we
ould take L ould to be a redu ed multi-variable p-adi L-fun tion.
For simpli ity, we now assume that J itself has omplex
multipli ation so that ' Z 2pg and
' Z p [[T1 ; : : : ; T2g ℄℄:
g be the hara ters of G
Let fi g2i=1
F;T appearing in Tp J and
i = i .
32
The hara ters that appear in [W i+1 nW i ℄ are all of the form
j1 j2 j3 ji ;
where j1 < j2 j3 ji ; ea h with multipli ity at most one.
For su h a hara ter to ontribute to Hom (M ( 1); [W i+1 nW i ℄ ),
we must have
j1 j2 j3 ji (L) = 0:
Furthermore, for ea h su h hara ter, 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 redu es to ounting the number of zeros of L
among su h hara ters.
33
The bulk of the ontribution omes from indi es 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 de reasing sequen es (j3 ; : : : ; ji ) of numbers from f1; 2; : : : ; 2gg.
34
When we try to bound
X dimH (G ; W
n
i=2
2
T
i+1 nW i );
the possible multi-indi es as i goes from 2 to n run over the latti e
points inside a simplex of edge length n 2 inside a 2g-dimensional
spa e. Using a hange of variable one an always redu e 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 dedu es a bound O(n2g 1 ) for the
number of zeros.
2
35
Remark:
Finiteness for ellipti
urves follows the pattern
Non-vanishing of L-fun tion ) niteness of Selmer group
) niteness of points.
For urves of higher genus with CM Ja obians, the impli ations are
Sparseness of L-zeros
niteness of points.
) bounds for Selmer varieties )
36