The priniple of Birh and Swinnerton-dyer for
ertain hyperboli urves
Minhyong Kim
November 12, 2007
London-Paris Number Theory Seminar
1
I. Preliminary remarks
II. Arithmeti fundamental groups
III. Selmer varieties
IV. Ellipti urves with omplex multipliation
V. Preliminary remarks
2
I. Preliminary remarks
(i) Path torsors
M reasonable onneted topologial spae. b 2 M , point.
Determines a fundamental group:
1 (M; b):
More generally, onsider b; x 2 M , and the set
1 (M ; b; x)
of homotopy lasses of paths from b to x.
Wish to study dependene on variable x.
3
Somewhat more struture:
Eah
has an ation of
1 (M ; b; x)
1 (M; b)
on the right
1 (M ; b; x) 1 (M; b)!1 (M ; b; x)
(p; ) 7! p Æ turning it into a torsor for 1 (M; b).
We have a variation of torsors.
4
In set-theoreti setting, problem trivial. Whenever we hoose an
element p 2 1 (M ; b; x), ation indues a bijetion
1 (M; b) ' 1 (M ; b; x)
7! p Æ That is to say, the hoie of any element determines a trivialization.
But suh isomorphisms are not anonial, as is often emphasized in
elementary topology. That is, any two torsors 1 (M ; b; x1 ) and
1 (M ; b; x2 ) are isomorphi, but not anonially. Usually, this
distintion is not important.
However, geometrially, these torsors form a natural family.
5
(ii) Covering spaes
Reall another interpretation of the fundamental group. Let
M~ !M
be a universal overing. Fix a point ~b 2 M~ so that we get a map
~ ~b)!(M; b)
(M;
of pointed spaes.
This map is universal among pointed overing spaes.
6
That is, given any pointed overing spae
(Y; y)!(M; b)
there is a unique ommutative diagram
~ ~b)
- (Y; y)
(M;
-
?
(M; b)
7
~ ~b), we get an isomorphism
Using (M;
1 (M ; b; x) ' M~ x
by lifting paths. Thus, the study of the variation of 1 (M ; b; x) in
x, beomes that of studying the bers of
M~ !M
This ber bundle is of ourse not trivial in general, i.e., the
universal 1 (M; b)-torsor is not trivial.
[One onstrution of M~ onsists in
M~ := [x (M ; b; x)℄
8
Of ourse we are interested in the situation where
X=Q
is a variety,
M = X (C )
and
b; x 2 X (Q ) M
That is, in 1 (X (C ); b; x) for speial points x.
But in this form, an't distinguish suh speial points from generi
ones.
9
II. Arithmeti fundamental groups
(i) Galois Ations
Consider the pronite ompletion
1 (X (C ); b)^
and the pushout torsor
1 (X (C ); b; x)^ = 1 (X (C ); b; x) 1 (X (C );b) 1 (X (C ); b)^
10
Then 1 (X (C ); b)^ and 1 (X (C ); b; x)^ end up with ompatible
ations of
:= Gal(Q =Q ):
Compatibility means that for g 2 , l 2 1 (X (C ); b)^ and
p 2 1 (X (C ); b; x)^ , then
g(p)g(l) = g(pl)
Thus, 1 (X (C ); b; x)^ beomes a -equivariant torsor for
^1 (X (C ); b).
[Or a torsor on the etale site of Spe(Q ).℄
11
Underlying this ation are the isomorphisms
b)
1 (X (C ); b)^ ' ^1 (X;
and
1 (X (C ); b; x)^ ' ^1 (X ; b; x)
involving the pronite etale fundamental group and the etale torsor
of paths for
X := X Spe(Q ) Spe(Q )
12
Dened using Cov(X ), the ategory of nite etale overing spaes
of X and the ber funtors
Fb : Cov(X )!nite sets
Y
Yb
X
b
# 7! #
13
Funtorial denition:
b) := Aut(Fb )
^1 (X;
^1 (X ; b; x) := Isom(Fb ; Fx )
Then ats on the ategory preserving the ber funtors (when
b; x 2 X (Q )) and hene ats on the group and torsor.
[It is this denition that allows us to study exibly the base-point
dependene.℄
14
(ii) Universal pro-overing spaes
To ompute this ation, again use a universal pointed overing
spae
~ ~b)!(X;
b)
(X;
onstruted, for example, using Galois theory, with the universal
property that given any nite algebrai overing spae
b) there is a unique ommutative diagram
(Y; y)!(X;
~ ~b)
- (Y; y)
(X;
-
?
(X; b)
15
~ ~b) is atually a projetive system
(X;
f(Xi; bi)g
and the diagram means that there is some index i and a
ommutative diagram
(X i ; bi ) - (Y; y)
-
?
(X; b)
In this situation, one again, we have
b) ' X~ b
^1 (X;
and the path spae
^1 (X ; b; x) ' X~ x
16
As the notation suggests, an take the whole system X~ , i.e., eah
X i !X;
the transition maps between them, and the base point ~b = fbi g to
be dened over Q . And then the Galois ation just beomes the
naive ation on the bers of
X~ !X
over rational points.
17
Example:
0) ellipti urve with origin over Q . Let
(E;
En !E
be the overing spae given by E itself with the multipliation map
[n℄ : E !E
Then the system
~ ~0) := f(En ; 0)gn
(E;
- (E; 0)
is a universal pointed overing spae.
Thus, for (E; 0),
0) ' T^(E )
^1 (E;
and an element of the fundamental group is just a ompatible
olletion of torsion points of E .
18
Similarly,
^1 (E ; 0; x) ' E~x
onsists of ompatible systems of division points of x.
This example illustrates that if we take into aount the Galois
ation, it is no longer possible to trivialize the torsor in general,
even point-wise.
b) and
That is, there will usually be no isomorphism between ^1 (X;
^1 (X ; b; x) in the ategory of -equivariant torsors. [Or as sheaves
on Spe(Q ).℄
19
In the ase of (E; 0), if there were an isomorphism
0) ' ^1 (E ; 0; x)
^1 (E;
then there would be a Galois invariant element of
^1 (E ; 0; x) ' E~x :
In partiular, for any n, there would be a rational point xn suh
that nxn = x. Not possible by Mordell's theorem.
[In general, a -equivariant torsor an be trivialized if and only if it
has a -invariant element.℄
20
(iii) The general formalism of arithmeti period maps
b) hoose any element
Given a -equivariant torsor T for ^1 (X;
t 2 T . Then for eah g 2 , g(t) is related to t by the
b)-ation, i.e.,
^1 (X;
g(t) = tg
b). The map g 7! g obtained thereby
for some g 2 ^1 (X;
determines a non-abelian (ontinuous) oyle
:
!^1 (X ; b; x);
that is, satisfying the relation
(g1 g2 ) = (g1 )g1 ((g2 ))
21
The set of suh oyles is denoted by
b))
Z 1 ( ; ^1 (X;
b) itself ats on the set of oyles via
^1 (X;
()(g) = g( 1 )(g)
giving rise to the set of orbits
b)) := ^1 (X;
b)nZ 1 ( ; ^1 (X;
b))
H 1 ( ; ^1 (X;
This is a non-abelian ohomology set lassifying the -equivariant
b).
torsors for ^1 (X;
22
Thus, the previous disussion of varying torsors of paths an be
summarized as a `period' map
b))
X (Q )!H 1 ( ; ^1 (X;
x 7! [^1 (X ; b; x)℄
Suppose X is a ompat smooth urve of genus 2. Then this map
is injetive by the Mordell-Weil theorem.
Grothendiek's setion onjeture
proposes that this map is surjetive as well, i.e., torsors of paths
oming from rational points are the only natural torsors for
b). Part of his anabelian program.
^1 (X;
23
(iv) The Diophantine onnetion
Grothendiek expeted setion onjeture to lead to another proof
of Diophantine niteness for hyperboli urves. Somewhat explains
his disapproval of motives? Theory of motives, involving
abelianization, rarely gives information on X (Q ). Serious deieny
indiating need to move beyond abelian motives.
b) is
However, orretness of expetation unlear. Perhaps ^1 (X;
too non-abelian. Need to nd middle ground between anabelian
and motivi language, or between hyperboli and ellipti urves.
24
For ellipti urves, the orresponding map
E (Q )!H 1 ( ; T^(E ))
is lassial, and its study is Kummer theory. In the theory of
ellipti urves, one onstruts a natural subspae
Hf1 ( ; T^(E )) H 1 ( ; T^(E ))
using loal onditions and onjetures that
\
E
(Q ) ' Hf1 ( ; T^(E ))
(Birh and Swinnerton-Dyer)
From this perspetive, the setion onjeture is a natural
non-abelian generalization of BSD.
25
Parallel piture:
E (Q )
X (Q )
x
- H 1(
- H 1(
; T^(E ))
b))
; ^1 (X;
- [^1(; b; x)℄
26
Finiteness then should follow from a kind of non-abelian BSD
priniple:
Non-vanishing of L-values ) Diophantine niteness.
Mostly speulative...
27
III. Selmer varieties
(i) Summary
This idea an be implemented for
-hyperboli urves of genus zero;
-the `Coates-Wiles' situation:
X = E n f0g
where E=Q is an ellipti urves with omplex multipliation;
-a few more sattered ases
using Selmer varieties.
28
Also, niteness for a general hyperboli urve follows from a `higher
BSD onjeture' suh as the Bloh-Kato onjeture, or the
Fontaine-Mazur onjeture. Both of these are assertions of
surjetivity of [e.g. regulator℄ maps from motives to some Hf1 ( ; ).
That is to say, so far, general niteness for urves aounted for by
abelian surjetivity + mildly non-abelian onstrution
29
(ii) Motivi fundamental groups
Fous now on a hyperboli urve X and the Q p -pro-unipotent
ompletion of its fundamental group.
where U Q
p
b)
^1 (X;
is dened using
-U
Qp
b)
= 1et;Q (X;
Un(X )Q
p
p
ategory of unipotent Q p -lisse sheaves of X .
[A sheaf is unipotent if it orresponds to a unipotent representation
b).℄
of ^1 (X;
30
Point b 2 X (Q ) again determines a linear ber funtor
Fb : Un(X )Q
and
p
!VetQ
p
U Q := Aut
(Fb )
p
For x 2 X (Q ) there is a torsor of unipotent paths
et;Q (X ; b; x) := Isom
(Fb ; Fx ) (' ^1 (X ; b; x) 1
Qp
)
U
^1 (X;b)
p
These objets also arry ompatible -ations.
31
The previous period map is replaed by
X (Q )
- H 1(
- [1et;
Qp
x
; UQ )
p
(X ; b; x)℄
Can study this indutively using the desending entral series
Z 1 := U Q
p
Z 2 := [U Q ; U Q ℄ Z 3 := [U Q ; [U Q ; U Q ℄℄ p
p
p
p
p
and the assoiated quotients UnQ := U Q =Z n+1 that t into exat
sequenes
0![Z n+1 nZ n ℄!UnQ !UnQ 1 !0
p
p
p
32
p
and indue a tower:
-
H 1 ( ; U4Q )
-
..
.
..
.
H 1 ( ; U3Q )
?
- H 1(
H 1(
X (Q )
lifting lassial Kummer theory.
33
p
?
; U2
?
p
Qp
)
; U1Q )
p
Can also onsider the loal ation of p := Gal(Q p =Q p ) and a loal
version of the tower
..
.
..
.
H 1 ( p ; U4Q )
-
-
?
H 1 ( p ; U3Q )
- H 1(
H 1(
X (Q p )
34
p
?
p ; U2
?
p
Qp
)
Qp
)
p ; U1
leading to a sequene of ommutative diagrams
X (Q )
#
H 1 ( ; Unet )
!
X (Q p )
#
! H 1 ( p; Unet)
(iii) Selmer varieties
We will utilize this diagram by way of a bit more geometri
information. Let S be a nite set of primes, Z S the ring of
S -integers, and
X!Spe(Z S )
a good model of X . [A smooth model with a smooth
ompatiation having an etale ompatiation divisor (possibly
empty).℄
Choose p 2= S and put T = S [ fpg.
35
Then we get an indued diagram:
X (Z S )
#glob
n
Hf1 ( ; Unet )
!
X (Z p )
#lo
n
! Hf1 ( p; Unet)
lo
where
Hf1 ( p ; Unet ) H 1 ( p ; Unet )
lassies torsors that are rystalline, i.e., have a p -invariant
Br -point, and
Hf1 ( ; Unet ) H 1 ( ; Unet )
lassies torsors that are unramied outside T and rystalline at p:
These notions allow us to fous the general formalism.
36
Two key points:
I. The loalizations
Hf1 ( ; Unet )
- Hf1(
et
p ; Un )
are maps of algebrai varieties over Q p .
In partiular,
-Hf1 ( ; Unet ) and Hf1 ( p ; Unet ) are natural geometri families into
whih the points t:
global and loal Selmer varieties;
-and the diÆult inlusion X (Z S ) X (Z p ) is replaed by an
algebrai map.
37
II. The map
1
Q
lo
n : X (Z p )!Hf ( p ; Un )
an be omputed using non-abelian p-adi Hodge theory.
In fat, Hodge theory provides a ommutative diagram:
X (Z p )
kn dr=
r
lo
n
p
? et
Hf1 ( p ; Un )
D
-
-
UnDR =F 0
and the map dr=r
an be expliitly omputed using p-adi
n
iterated integrals.
38
(iv) The loal map
Here
U DR = 1DR (X Q p ; b)
is the De Rham/rystalline fundamental group of X Q p and F i
refers to the Hodge ltration.
Then U DR =F 0 beomes a lassifying spae for De Rham/rystalline
torsors, and the map
X (Z p )!U DR=F 0
again assoiates to a point x the torsor
1DR (X Q p ; b; x)
of De Rham/rystalline paths.
39
Example:
X = P1 n f0; 1; 1g. Then the oordinate ring of U DR is the
Q p -vetor spae
Q p [w ℄
where w runs over words on two letters A; B . Also, F 0 = 0, and for
w = Am1 BAm2 B Am B
l
we get
=
Zx
w Æ dr=r (x)
(dz=z )m1 (dz=(1 z ))(dz=z )m2 (dz=z )m (dz=(1 z ))
l
b
a p adi multiple polylogarithm.
40
The map
D : Hf1 ( p ; UnQ )!UnDR =F 0
p
is given by
D(P ) = Spe([P Br ℄G )
if P = Spe(P ). Commutativity of the diagram is the assertion
p
1et;Q (X ; b; x) Br ' 1DR (X Q p ; b; x) Br
p
proved by Shiho, Vologodsky, Faltings, Olsson.
41
A orollary of this desription is that
Theorem 0.1
The image of eah
X (Z p )!Hf1( p; UnQ )
p
is Zariski dense. In fat, the image of eah residue disk is
Zariski-dense.
A poor man's loal substitute for the setion onjeture.
42
(v) Finiteness
Another orollary:
Theorem 0.2
Suppose
Im[Hf1 ( ; UnQ )℄ Hf1 ( p ; UnQ )
p
is not Zariski dense. Then
p
X (ZS ) is nite.
43
Idea of proof:
- X (Zp )
X (Z S ) glob
n
?
lo
n
Hf1 ( ; UnQ )
p
lo
-
?
Hf1 ( p ; UnQ )
p
9 6= 0
?
Qp
44
suh that vanishes on Im[Hf1 ( ; UnQ )℄. Hene, Æ lo
n vanishes
on X (Z S ). But this funtion is a non-vanishing onvergent power
series on eah residue disk. 2
p
45
(vi) Connetion to Iwasawa theory
In all ases so far, prove non-denseness by showing
dimHf1 ( ; UnQ ) < dimHf1 ( p ; UnQ )
for n >> 0. Implied by standard motivi onjetures.
That is, ontrolling the Selmer variety leads to niteness of points.
Nature of the inequality suggests that proofs should go through
Iwasawa theory.
p
p
46
Preliminary outline:
Hf1 ( ; UnQ ) H 1 ( T ; UnQ )
p
and
p
0!H 1 ( T ; Z n+1 nZ n )!H 1 (
T ; Un p )
Q
!H 1 ( T ; UnQ 1)
p
is an exat sequene.
Euler harateristi formula:
dimH 1 (
T;Z
n+1
nZ n ) dimH 2( T ; Z n+1nZ n ) = dim(Z n+1nZ n )
Therefore, ontrolling
H 2 ( T ; Z n+1 nZ n )
gives a bound on the dimensions of global Selmer varieties.
Meanwhile, dimension of loal Selmer varieties given by preise
ombinatorial formula.
47
For X = P1 n f0; 1; 1g,
Z n+1 nZ n ' Q p (n)r
n
and
H 2 ( T ; Z n+1 nZ n ) = 0
for n >> 0 follows from the niteness of zeros of p-adi L-funtion
for ylotomi Z p -extension of Q (p ).
[Can also use Soule's map from K-theory.℄
48
IV. Ellipti urves with omplex multipliation
For X = E n f0g, E ellipti urve with CM by an imaginary
quadrati eld K , need to hoose p to be split as p = in K and
replae U Q by a natural quotient W with property that
p
U2Q
p
' W2
and (for n 3)
W n =W n+1 ' Q p (
n 2 )(1)
Q p ( n 2)(1)
viewed as a representation of in the natural way, where
are haraters of N := Gal(Q =K ) orresponding to
T E := lim E [n ℄ and T E := lim E [n ℄ respetively.
49
and Notation:
s = jS j 1
r = dimHf1 ( ; U1Q )
M = K (E [1 ℄); M = K (E [1 ℄)
)
G = Gal(M=K ); G = Gal(M=K
= Z p [[G℄℄; = Z p [[G ℄℄
: !Q p dened by ation of G on T (E )
: !Q p dened by ation of G on T (E )
Vp = Tp (E ) Q , V , et.
Have orresponding p-adi L-funtions:
p
Lp 2 ; Lp 2 50
W:
= N < >, where is omplex onjugation.
Choose a Q p -basis e of T (E ) Q p so that f := (e) is a Q p -basis
of T (E ) Q p .
Reall that
U := LieU
an be realized as the primitive elements in
Constrution of
T (U1 ) = T (Vp )
where T ( ) refers to the tensor algebra (but with a dierent
Galois ation).
51
For example, if 2 N , then
[e; [e; f ℄℄ = ( )2 ( )[e; [e; f ℄℄ + Lie monomials of higher degree
and
[e; [e; f ℄℄ = [f; [f; e℄℄ + Lie monomials of higher degree
That is, U has a bi-grading
U = i;j1Ui;j
orresponding to e and f degrees, but whih is not preserved by the
Galois ation.
52
However, easy to hek that the ltration
Un;m := in;jmUi;j
is preserved by N , while
(Un;m ) = Um;n
So
Un;n
is Galois invariant for eah n.
Furthermore, it is a Lie ideal.
53
Hene, there is a well-dened quotient W of U orresponding to
U =U2;2
We then see that
W n =W n+1
'< ad(e)n 1(f ) > < ad(f )n 1(e) > (mod W n+1)
' n 2 (1) n 2(1)
54
Extended diagram:
X (Z S )
#
,!
X (Z p )
#
! Hf1( p; UnQ )
#
#
Hf1 ( ; Wn ) ! Hf1 ( p ; Wn )
Hf1 ( ; Un )
p
55
Theorem 0.3
dimHf1 ( ; Wn ) < dimHf1 ( p ; Wn )
for n >> 0.
56
Theorem 0.4
(*)
k(
Then
Assume
Lp) 6= 0 and k (Lp) 6= 0 for all k > 0.
dimHf1 ( ; Wn ) < dimHf1 ( p ; Wn )
for n = r + s.
Finiteness follows from the previous argument applied to these
modied Selmer varieties.
57
Proof of theorems
Uses main onjeture for K . We will onentrate on (0.4).
We need the exat sequene
0!W n =W n+1 !Wn !Wn 1 !0
As for the Hodge ltration,
dimW1DR =F 0 = 1
and
for n 2, so that
F 0 [(W DR )n =(W DR )n+1 ℄ = 0
dimHf1 ( p ; Wn ) = 2 + 2(n 2) = 2n 2
for n 2.
58
Meanwhile,
dimHf1 ( ; W1 ) = r
dimHf1 ( ; W 1 =W 2 ) = dimHf1 ( ; Q p (1)) = s 1
so that
dimHf1 ( ; W2 ) r + s 1
As we go down the lower entral series, we have, in any ase, the
Euler harateristi formula
dimH 1 ( T ; W n =W n+1 ) dimH 2 (
T;W
n =W n+1 )
= dim(W n =W n+1 )= 1 = 1
and
Hf1 ( ; W n =W n+1 ) = H 1 ( T ; W n =W n+1 )
for n 2, so we need to ompute the H 2 term.
59
Claim (still assuming (*)):
H 2 ( T ; W n =W n+1 ) = 0
for n 3.
Clearly, it suÆes to prove this after restriting to NT
obvious notation). Then we have
W n =W n+1 '
We will show
for n 3.
H 2 (NT ;
n 2 (1)
n 2 (1)
n 2 (1)) = 0
60
T
(with
Consider the loalization sequene
0!Sha2T (
!H 2(NT ;
n 2 (1)),
! vjT H 2(Nv ;
n 2 (1))
that denes the vetor spae Sha2 (
H 2 (Nv ;
n
n 2 (1)).
n 2 (1))
By loal duality,
2 (1)) ' H 0 (N ; 2 n ) = 0
v
sine the representation 2
potentially rystalline.
n
is potentially unramied or
61
So we have
H 2 (NT ;
n 2 (1))
' Sha2T (
n 2 (1))
' Sha1T (
2 n )
by Poitou-Tate duality. But
Sha1T ( 2 n ) ' Hom (A; 2 n )
where A is the Galois group of the maximal abelian unramied
pro-p extension of M (= K (E [1 ℄)) split above the primes dividing
T.
62
In partiular, A is annihilated by Lp .
Sine we are assuming 2 n (Lp ) 6= 0 for n 3, we get the desired
vanishing:
H 2 (NT ; n 2 (1)) = 0
Similarly,
H 2 (NT ; n 2 (1)) = 0
Finally, we onlude that
dimHf1 ( ; W n =W n+1 ) = 1
for n 3 so that
dimHf1 ( ; Wn ) r + s + n 3
for n 2.
63
Thus,
Hf1 ( p ; Wn ) = 2n 2 > r + s + n 3 = dimHf1 ( ; Wn )
as soon as n r + s.
Note that even without (*), we have
2 n (L ) 6= 0 2 n (L ) 6= 0
p
p
and hene,
H 2 ( T ; W n =W n+1 ) = 0
for n suÆiently large.
64
Therefore,
dimHf1 ( ; Wn ) n < dimHf1 ( p ; Wn ) 2n
for n suÆiently large, yielding niteness of
X (ZS )
in any ase.
However, the eetivity in n that appears in (0.4) should eventually
apply to the problem of nding points.
65
V. Preliminary remarks
Non-abelian priniple of Birh and Swinnerton-Dyer:
non-vanishing of (most) L-values ) ontrol of Selmer
varieties ) niteness of integral points
in parallel to the ase of ellipti urves, with just the substitution
of Selmer varieties for Selmer groups. But the ases studied so far
should just be a shadow of the true piture, where, for example, the
non-vanishing of L should be a non-abelian statment.
Also, both impliations should eventually be diret.
66
Relevane of setion onjeture: omplete omputation of points
and non-abelian desent.
67