Selmer varieties
Minhyong Kim
February, 2009
Regensburg
1
I. Non-abelian desent
II. Diophantine niteness
III. Preliminary remarks on non-abelian duality
IV. Expliit formulas.
2
General omment:
Usual approah to the Diophantine geometry of urves emphasizes
the dierenes between three ases: genus zero, genus one, and
genus 2.
However, we wish to fous on the parallels, espeially between
(E; e);
a ompat urve of genus one equipped with an integral point, and
(X; b);
a hyperboli urvea equipped with an integral point.
a That
C)
is, X (
has a hyperboli metri. Equivalently, X is genus zero minus
at least three points, genus one minus at least one point, or genus
3
2
.
I. Non-abelian desent
(E; e) ellipti urve over Z S . p prime not in S . T = S [ fpg.
G := Gal(Q =Q ). Kummer theory provides an injetion
E (Z S )=pn E (Z S ),!H 1 (GT ; E [P n ℄):
BSD onjetured an isomorphism
E (Z S ) Z p ' Hf;1 Z(G; Tp (E ))
where
Tp (E ) := lim E [pn ℄
is the p-adi Tate module of E and the subsript f; Z refers to loal
`Selmer' onditions.
4
When X=Z S is a smooth hyperboli urve and b 2 X (Z S ), analogue
of above onstrution is
1
Z p ))
X (Z S ) !
Hf (G; H1et (X;
using the p-adi étale homology
Z p ) := et;p (X;
b)ab
H1et (X;
1
of X := X Spe(Q ) Spe(Q ).
Several dierent desriptions of this map.
5
But in any ase, it fators through the Jaobian
X (Z S )!J (Z S )!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 (Z S ) from J (Z S ).
Eorts of Weil, Mumford, Vojta.
6
The theory of Selmer varieties renes this to a tower:
Hf1 (G; U4 )
-
?
Hf1 (G; U3 )
?
- Hf1(G; U2)
?
- 1
3 4
-
..
.
..
.
X (Z S )
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
7
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 .
8
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
U1 = H1et (X;
:= Tp J
Qp
at the bottom level and exat sequenes
0!U n+1 nU n !Un !Un 1 !0
for eah n.
9
For example, for n = 2,
0! ^2 V !U2 !V !0;
for ane X .
When X = E n feg for an ellipti urve E , this beomes
0!Q p (1)!U2 !V !0:
When X is ompat, we get
0![^2 V=Q p (1)℄!U2 !V !0;
where
Q p (1),! ^2 V
omes from the Weil pairing.
10
4. U has a natural ation of G lifting the ation on V , and
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.
11
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 ).
12
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.)
13
7. The map
na = fn g : X (Z S ) - Hf1 (G; U )
is dened by assoiating to a point x the prinipal U -bundle
P (x) = u;Q (X ; b; x) := Isom
(Fb ; Fx )
1
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 (Z S )!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, suggests the
possibility of separating the struture of X (Z S ) from that of J (Z S ).
14
8. If one restrits U to the étale site of Q p , there are loal analogues
1
na
p : X (Z 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.
15
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 (Z p )
- Hf1(Gp; U )
na
p
d na
r
=
r
D
- ?
DR 0
U
=F
whose ommutativity redues to the assertion that
1DR (X ; b; x) Br ' 1u;Q (X ; b; x) Br :
p
16
9. The map
DR =F 0
na
:
X
(
Z
)
!
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 (Z 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.
17
10. The loal and global onstrutions t into a family of
ommutative diagrams
- X (Zp )
X (Z S )
?
H 1 (G; U
f
n)
?
-
- Hf1(Gp; Un) -D UnDR=F 0
lo
p
where the bottom horizontal maps are algebrai, while the vertial
maps are transendental. Thus, the diult inlusion
X (Z S ) X (Z p ) has been replaed by the algebrai map lop .
18
II. Diophantine Finiteness
Theorem 1
Suppose
lop (Hf1 (G; Un )) Hf1 (Gp ; Un )
is not Zariski dense for some n. Then X (Z S ) is nite.
Theorem is a rude appliation of the methodology. Eventually
would like rened desriptions of the image of the global Selmer
variety, and hene, of X (Z S ) X (Z p ) by extending the method of
Chabauty and Coleman and the work of Coates-Wiles, Kolyvagin,
Rubin, Kato on the onjeture of Birh and Swinnerton-Dyer.
19
Idea of proof: There is a non-zero algebrai funtion - X (Zp )
X (Z S ) na
n
?
Hf1 (G; Un )
na
p;n
?
- Hf1(Gp; Un)
DÆlop
96=0
?
Qp
vanishing on lop [Hf1 (G; Un )℄. Hene, Æ na
p;n vanishes on X (Z S ).
But using the omparison with the De Rham realization, we see
that this funtion is a non-vanishing onvergent power series on
eah residue disk. 2
20
-Hypothesis of the theorem expeted to always hold for n
suiently large, but diult to prove. For example, Bloh-Kato
onjeture on surjetivity of p-adi Chern lass map, or
Fontaine-Mazur onjeture on representations of geometri origin
all imply the hypothesis for n >> 0.
That is, Grothendiek expeted
Non-abelian `niteness of
niteness of X (Z S ).
X' (= setion onjeture) )
Instead we have:
`Higher abelian niteness of
X' ) niteness of X (ZS ).
21
Can prove the hypothesis in ases where the image of G inside
Z p )) is essentially abelian. That is, when
Aut(H1 (X;
-X has genus zero;
-X = E n feg where E is an ellipti urve with omplex
multipliation;
-(with John Coates) X ompat of genus 2 and JX fators into
abelian varieties with omplex multipliation. For example, X
might be
for n 4.
axn + byn = z n ;
In the CM ases, need to hoose p to split inside the CM elds.
22
Idea: Construt a quotient
U !W !0
and a diagram
- X (Z p )
X (Z S ) 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
23
?
- WnDR=F 0
D
suh that
dimHf1 (G; Wn ) < dimWnDR =F 0
for n >> 0.
When JX has CM, an onstrut a `polylogarithmi quotient'a
W
= U=[[U; U ℄; [U; U ℄℄
suh that all CM haraters
i1 i2 i
n
appearing in W n =W n+1 have multipliity one.
a The
terminology is adopted by analogy with the quotient of the fundamen-
tal group of
P1 n f0 1 1g
;
;
that provides the natural setting for the theory of
polylogarithms aording to Beilinson and Deligne.
24
Elementary Iwasawa theory an be used to show that
H 2 (GT ; i1 i2 i
n
)=0
rather generially, allowing us to ontrol
dimHf1 (G; Wn )
X
dimH
n
i=1
X
dimH
n
i=1
1
i =W i+1 ):
(
G;
W
f
1
(GT ; W i =W i+1 )
and show that it grows more slowly than
dimWnDR =F 0 :
25
III. Preliminary remarks on non-abelian duality
As far as the arithmeti of urves is onerned one major goal of
the theory is to give an expliit desription of
X (Z S ) X (Z p ):
This should ome from a non-abelian loal-global duality together
with a non-abelian expliit reiproity law.a Wish to provide a hint
of this idea using a very speial situation:
X = E n feg;
where EQ is an ellipti urve suh that
L(EQ ; 1) = 0; L0 (EQ ; 1) 6= 0:
Here, we will assume that E is in fat a minimal Z -model of EQ
and X = E n feg.
a Of
ourse both notions are highly speulative at present.
26
Choose S so that ES := E Spe(Z) Spe(Z S ) is smooth. Our
assumptions imply that
is nite and
X(E)
rankE (Z ) = rankE (Z S ) = rankE (Q ) = 1;
but still diult to analyze the inlusion
X (Z ) E (Z ):
Some arhimedean progress using Diophantine approximation
supplemented by LLL-algorithm. Here, we fous on
non-arhimedean tehniques and the inlusion
X (Z ) X (Z p ):
27
Pik a tangential base-point b 2 Te E and only use U2 , identied
with its Lie algebra L2 . L2 has a natural graded struture
L2 = L[1℄ L[2℄
suh that L[1℄ ' L1 ' V (E ) and L[2℄ ' Q p (1) is the
one-dimensional enter of L2 . The group law on U2 an be thought
of as a twisted operation on L2 given bya
(l1 + l2 ) (l10 + l20 ) = l1 + l10 + l2 + l20 + (1=2)[l1 ; l10 ℄:
The grading is ompatible with the Galois ation (in fat, with the
motivi struture):
g(l1 + l2 ) = g(l1 ) + g(l2 ):
a This
kind of struture is often referred to as a
28
Heisenberg group
.
If a is a ohain on GT or Gp with values in U2 , then we an write
a = a1 + a2
with a1 taking values in L[1℄ and a2 taking values in L[2℄. The
oyle ondition for the group law is written in terms of these
omponents as
da1 = 0
da2 =
(1=2)a1 [ a1 ;
where the up produt of two ohains and is dened by
[ (g; h) = [ (g); g(h)℄:
29
We will onstrut a funtion on Hf1 (Gp ; U2 ) using seondary
ohomologial operations.
Let : GT !Q p be the log of the p-adi ylotomi harater and
p = jGp . Given a oyle = 1 + 2 : Gp !U2 , the funtion is, in
essene,
7! (; 1 ; 1 );
where the braket refers to a Massey triple produt, taking values in
H 2 (Gp ; L[2℄) ' Q p :
This notion omes from rational homotopy theory, and is usually
dened for ohomology lasses of an assoiative dierential graded
algebra A.
30
If
[a℄ 2 H 1 (A);
[b℄ 2 H 1 (A);
[℄ 2 H 1 (A)
are lasses with the property that
[a℄[b℄ = 0;
[b℄[℄ = 0;
then we an solve the equations
dx = ab; dy = b:
We see then that
x + ay
is a oyle, dening a lass in H 2 (A). Note that the lass depends
on the hoie of x and y, a dening system. Well-dened lass lives
only inside
H 2 (A)=[aH 1 (A) + H 1 (A)℄:
31
In our situation, the omplex of ohains on Gp with values in
Q p L[1℄ L[2℄
forms an assoiative dierential graded algebra. Given a oyle
= 1 + 2 : Gp !U2 ;
we have
[p ℄ [ [1 ℄ = 0;
sine H 2 (Gp ; L[1℄) = 0: Also,
sine d( 22 ) = 1 [ 1 :
[1 ℄ [ [1 ℄ = 0;
Therefore, we an form the Massey triple produt
(p ; 1 ; 1 ) 2 H 2 (Gp ; L[2℄)=[p [ H 1 (Gp ; L[1℄)) + 1 [ H 1 (Gp ; L[1℄℄:
Unfortunately, zero.
32
But note that this naive Massey produt does not use the full data
of or the strength of our assumptions. Firstly, part of a dening
system for the Massey produt is enoded in :
d(
22 ) = 1 [ 1 :
Seondly, if [ ℄ 2 Hf1 (Gp ; U2 ), then [1 ℄ 2 Hf1 (Gp ; L[1℄) is in the
image of the loalization map
Hf;1 Z(G; L[1℄) ' Hf1 (Gp ; L[1℄):
Hene, the equation
dx = [ 1
makes sense globally.
33
Key point:
Using Using our restritive hypotheses, there is a global solution
xglob : GT !L[1℄
to the equation
Uses the niteness of
dx = [ 1 :
X and a generator for E(Z).
34
Proposition 2
The lass
glob
p
2
p ( ) := [lop (x ) [ 1 + [ ( 22 )℄ 2 H (Gp ; L[2℄)
is independent of all hoies.
Thus,
1
p : Hf (Gp ; U2 )!Q p
is a well-dened algebrai funtion on the loal Selmer variety.
Remark: The map
Z p Q p
' Hf1(Gp ; L[2℄),!Hf1(Gp ; U2) ! Q p
p
is the log map, and hene, p is non-zero.
35
Dene (`rened Selmer variety')
Hf;1 Z(G; U2 ) Hf1 (G; U2 )
to be the intersetion of the kernels of
lol : Hf1 (G; U2 )!Hf1 (Gl ; U2 )
for all l 6= p.
In fat, we have a ommutative diagram
- X (ZS )
X (Z ) ?
?
Hf;1 Z(G; U2 ) - Hf1 (G; U2 )
Joint work with A. Tamagawa.
36
Theorem 3 (loal-global duality)
Hf;0 (G; U2 )
1
lop
The map
! Hf1(Gp ; U2) ! Q p
p
is zero.
Proof is straightforward using the standard the exat sequene
0!H 2 (GT ; Q p (1),! v2T
37
H 2 (Gv ; Q p (1))!Q p !0:
IV. Expliit formulas
Choose a Weierstrass equation for E and let
= dx=y; = xdx=y:
Dene
log (z ) :=
Z
z
b
;
log (z ) :=
D2 (z ) :=
Z
via (iterated) Coleman integration.
38
b
z
;
Z
z
b
;
Corollary 4
Then the set
Suppose we have a point y 2 X (Z ) of innite order.
X (Z ) X (Z p )
lies inside the zero set of the analyti funtion
log2 (y ))(D2 (z )
Atually,
[D2 (z )
log (z ) log (z ))
pÆD
1
log (y ) log (y )):
Æ na
DR=r;2 = Rese (vdx=y )
log (z ) log (z )
where dv = xdx=y .
log2 (z )(D2 (y )
log (z ) 2
(
) (D2 (y )
log (y )
39
log (y ) log (y ))℄;
Remark:
-In partiular, the funtion
(D2 (z )
log (z ) log (z ))=(log (z ))2
is onstant on the integral points of innite order.
-Parts of this onstrution generalize to ane urves X of genus
g 2 whose Jaobians have Mordell-Weil rank g.
-Also to ompat urves X provided
rankNS (JX ) 2:
! Possibility of omputing points on urves of genus 2 with rank 2
Jaobians.
40