A non-abelian priniple of Birh and
Swinnerton-Dyer
Minhyong Kim
Muenster
24 June, 2008
1
I. Non-abelian Jaobians
Z S : ring of S -integers for a nite set S of primes.
X=Z S : smooth hyperboli urve with good ompatiation.
Here, hyperboliity refers to the ondition that X (C ), the omplex
points of X , has a non-abelian fundamental group.
Fix b 2 X (Z S ).
2
Kummer theory:
X (Z S )
?
JX (Z S )
- H 1(
Z^ ))
; H1 (X;
'
- H 1(
?
; T^(JX ))
JX : (generalized) Jaobian of X .
= Gal(Q =Q ).
Unfortunate fat that map fators through the Jaobian. Defet of
the theory of motives.
3
Natural lift:
b))
H 1 ( ; 1 (X;
-
na
X (Z S )
- H 1(
where
?
Z^ ))
; H1 (X;
na (x) := [1 (X ; b; x)℄;
a non-abelian Kummer map.
4
Grothendiek's setion onjeture: For X ompat,
b)):
na : X (Z S ) ' H 1 ( ; 1 (X;
Endows the set of points X (Z S ) with a non-abelian struture.
DiÆult point is surjetivity, as in the abelian theory of ellipti
urves.
Grothendiek expeted appliation to niteness of X (Z S ).
5
Can think of non-abelian ohomology as a non-abelian Jaobian (in
a pro-nite etale realization).
Grothendiek's proposal (80's) is therefore to use a non-abelian
Jaobian to rene the study of Diophantine problems on
non-abelian spaes.
Idea arose earlier in the 1930's.
6
Weil:
Generalisation des fontions abeliennes
`A text presented as analysis, whose signiane is essentially
algebrai, whose motivation is arithmeti!' (Serre)
Paper is thought of as foundational in the theory of vetor bundles
on urves, leading to Narasimhan-Seshadri, Simpson, and a general
non-abelian Hodge theory.
Motivation in question was the Mordell onjeture a.k.a. Faltings'
theorem.
Weil thought to apply the `non-abelian Jaobian' Mn;0 (X ) to
Diophantine niteness, overoming the abelian deieny of the
Jaobian (or the ategory of motives).
7
Lak of natural Albanese map.
The moduli spae
b))
H 1 ( ; 1et (X;
on the other hand, omes equipped with the obvious Albanese map.
x 7! [1 (X ; b; x)℄
Nevertheless, appliation to niteness unlear.
8
For ellipti urves, one onjetures an isomorphism
\
E
(Q ) ' Hf1 ( ; T^(E )) H 1 ( ; T^(E ))
that applies to the desent algorithm.
Thus, Grothendiek's setion onjeture should be related to
non-abelian desent and an algorithm for nding points.
Finiteness itself should arise from a BSD priniple and
non-vanishing of L-values.
9
II. Motivi fundamental groups
A tratable moduli spae with geometri struture is obtained from
the motivi fundamental group.
Comprised of many omponents, as usual in the (onjetural)
theory of motives.
10
Of most diret use is the Q p -pro-unipotent etale fundamental group:
b)
U et = 1et;Q p (X;
dened using
Un(X )Q p ;
the ategory of unipotent loally onstant Q p -sheaves over X ,
following the general reipe.
[A sheaf is unipotent if van be ltered into onstant sheaves.℄
11
Point x 2 X (Q ) again determines a linear ber funtor
Fx : Un(X )Q p !VetQ p
and
U et := Aut
(Fb )
This has the struture of a pro-algebrai pro-unipotent group over
Q p.
For any x 2 X (Q ) there is a torsor of unipotent paths
et;Q p (X ; b; x) := Isom
(Fb ; Fx );
1
a pro-algebrai prinipal bundle for U et .
These objets are also sheaves on Spe(Q ).
12
The previous non-abelian Kummer map is then replaed by
X (Q )
- H 1(
; U et )
- [1et;Qp (X ; b; x)℄
x
that an be studied indutively using the desending entral series
Z 1 := U et Z 2 := [U et ; U et ℄ Z 3 := [U et ; [U et ; U et ℄℄ and the assoiated quotients Unet := U et =Z n+1 that t into exat
sequenes
0![Z n+1 nZ n ℄!Unet !Unet 1 !0
13
and indue a tower:
-
H 1 ( ; U4et )
-
..
.
..
.
H 1 ( ; U3 )
? et
? et
?
- H 1( ; U1et)= H 1(
H 1 ( ; U2 )
X (Q )
lifting lassial Kummer theory.
14
Q p ))
; H1et (X;
-
-
In the pro-unipotent theory, a loal version of the tower
..
.
..
.
H 1 ( p ; U4et )
X (Q p )
? et
H 1 ( p ; U3 )
- H 1(
- H 1(
? et
p ; U2 )
? et
p ; U1
)
with p = Gal(Q p =Q p ) arries onsiderable information.
15
Cruially, there is a sequene of ommutative diagrams
X (Q )
#
!
X (Q p )
#
H 1 ( ; Unet ) ! H 1 ( p ; Unet )
that an be rened using more geometri input.
By ontrast, strutured loal information seems inaessible in the
pro-nite theory.
16
III. Selmer Varieties
We go bak to S -integral points X (Z S ), hoose p 2= S , and put
T = S [ fpg.
Then we get a rened diagram:
- X (Z p )
X (Z S )
glob
n
Hf1 (
lo
n
? et lo- 1 ? et
Hf ( p ; Un )
; Un )
17
Note that the previous lassifying spaes have been replaed by
subsets
Hf1 ( p ; Unet ) H 1 ( p ; Unet )
and
Hf1 ( ; Unet ) H 1 ( ; Unet )
dened by niteness onditions reeting the integrality of the
points and algebrai geometri onstraints.
These are the loal and global Selmer Varieties: the extra
geometri struture on the Unet has endowed the moduli spaes also
with the struture of varieties.
18
The subset
Hf1 ( p ; Unet ) H 1 ( p ; Unet )
onsists of torsors that are rystalline, i.e., trivialized over
Fontaine's ring Br Q p of p-adi periods.
The global Selmer variety
Hf1 ( ; Unet ) H 1 ( ; Unet )
lassies torsors that are unramied outside T and rystalline at p.
i.e., the torsors that extend to Spe(Z T ) and are rystalline at p.
These renements allow us to fous the general formalism.
19
We now have a tower of `non-abelian Jaobians':
..
.
..
.
Hf1 ( ; U4et )
-
-
? et
Hf1 ( ; U3 )
- Hf1(
Hf1 (
X (Z S )
? et
; U2 )
? et
Q p ))
; U1 )= Hf1 ( ; H1et (X;
where the targets are now algebrai varieties.
20
Two further properties:
1. The loalizations
Hf1 ( ; Unet )
- Hf1(
p ; Unet )
are maps of algebrai varieties over Q p .
In partiular, the diÆult inlusion X (Z S ) X (Z p ) is replaed by
an algebrai map.
- X (Zp )
X (Z S ) glob
n
Hf1 (
lo
n
? et lo- 1 ? et
; Un )
Hf ( p ; Un )
21
2. The loal maps
1
et
lo
n : X (Z p )!Hf ( p ; Un )
an be omputed using non-abelian p-adi Hodge theory, in
partiular, the De Rham/rystalline fundamental group.
Thus, Hodge theory reappears.
22
IV. Analysis of the loal map
The De Rham fundamental group
U DR = 1DR (X; b);
dened using the ategory
UnDR (X )
of unipotent algebrai vetor bundles on X with at onnetion.
Fb : UnDR (X )!VetQ
(V; r) 7! Vb
U DR (X ) := Aut
(Fb )
1DR (X ; b; x) := Isom
(Fb ; Fx )
23
De Rham/rystalline strutures:
-Hodge ltration
1DR (X ; b; x) F i F i+1 F 0
-Ation of rystalline Frobenius
p : 1DR (X ; b; x) Q p !1DR (X ; b; x) Q p
oming from a omparison
1DR (X ; b; x) Q p ' 1DR (X Q p ; b; x)
' 1r (Y ; b; x)
where
Y = X Fp
24
In fat,
U DR (Q p )=F 0
is a lassifying spae for U DR Q p torsors with ompatible Hodge
ltration and Frobenius so that we also have a map
X (Z p )
- U DR(Q p )=F 0
kndr=r
that again assoiates to a point x the De Rham/rystalline torsor
of paths
1DR (X Q p ; b; x):
25
In fat, Hodge theory provides a ommutative diagram:
X (Z p )
lo
n
Hf1 (
kndr=
r
? et -' DR
Un (Q p )=F 0
p ; Un )
and the map dr=r
an be expliitly omputed using p-adi
n
iterated integrals.
26
Example:
X = P1 n f0; 1; 1g. Then the oordinate ring of U DR Q p 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 Aml B
we get
=
Zx
b
w Æ dr=r (x)
(dz=z )m1 (dz=(1 z ))(dz=z )m2 (dz=z )ml (dz=(1 z ))
a p adi multiple polylogarithm.
27
The isomorphism
D : Hf1 ( p ; Unet ) ' UnDR =F 0
is given by
D(P ) = Spe([P Br ℄Gp )
if P = Spe(P ). Commutativity of the diagram is the assertion
1et;Q p (X ; b; x) Br ' 1DR (X ; b; x) Br
proved by Shiho, Vologodsky, Faltings, Olsson.
28
An easy onsequene of this desription is that
Theorem 0.1
The image of
X (Z p )!Hf1 ( p ; Unet )
is Zariski dense. In fat, the image of eah residue disk in X (Z p ) is
Zariski-dense.
A poor man's loal substitute for Grothendiek's onjeture.
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Diophantine Appliations
Theorem 0.2
Suppose
Im[Hf1 ( ; Unet )℄ Hf1 ( p ; Unet )
is not Zariski dense. Then X (Z S ) is nite.
Idea of proof:
- X (Z p )
X (Z S ) glob
n
Hf1 (
lo
n
? et lo- 1 ? et
Hf ( p ; Un )
; Un )
96=0
?
Qp
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suh that vanishes on Im[Hf1 ( ; Unet )℄. Hene, Æ lo
n vanishes
on X (Z S ). But this funtion is a non-vanishing onvergent power
series on eah residue disk. 2
This proof for n = 1 is due to Chabauty.
Note that for niteness, we atually just need
Im(X (Z S )) Hf1 ( p ; Unet )
to be non-dense. One of many possible variations. In partiular,
will often end up studying
Im(X (Z S )) Hf1 ( ; Unet ):
31
V. Chabauty plus epsilon: Heisenberg groups
Suppose X is aÆne. Then we have an exat sequene
Q p )!U2et !U1et !0
0! ^2 H1 (X;
The Weil pairing gives a map
Q p )!Q p (1)
^2H1(X;
and, thereby, a pushout diagram:
0!
Q p ) ! U2et ! U1et !0
^2H1 (X;
#
#
#
0!
Q p (1)
! Het ! U1et !0
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We get an indued diagram
X (Z )
#
!
Hf1 ( ; Het )
Theorem 0.3
X (Z p )
#
! Hf1 ( p; Het )
Suppose dimJX (Z ) Q p g . Then
Im(X (Z )) ( Hf1 ( p ; Het ):
Remark: Works also for ompat X provided the Neron-Severi
group of JX has rank 2.
33
VI. (Philosophial) Connetion to Iwasawa theory
In many ases, prove non-denseness by showing
dimHf1 ( ; Unet ) < dimHf1 ( p ; Unet )
for n >> 0. Standard motivi onjetures imply that this should
hold in general. That is, in ontrast to the original method of
Chabauty, this method should always yield niteness.
Note that niteness is a onsequene of bounds on the Selmer
variety.
Nature of the inequality suggests that proofs should go through
Iwasawa theory.
34
VII. 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 et by a natural quotient W with property that
U2et ' 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.
35
and Notation:
s = jS j
r = dimHf1 ( ; U1et )
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:
Lp 2 ; Lp 2 36
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).
37
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.
38
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.
39
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)
40
Extended diagram:
X (Z S )
#
,!
X (Z p )
#
! Hf1( p ; Unet)
#
#
Hf1 ( ; Wn ) ! Hf1 ( p ; Wn )
Hf1 ( ; Un )
41
Theorem 0.4
dimHf1 ( ; Wn ) < dimHf1 ( p ; Wn )
for n >> 0.
42
Theorem 0.5
(*)
Assume
k (Lp ) 6= 0
Then
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.
43
Proof of theorem 0.5
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.
44
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.
45
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 ' n 2 (1) n 2 (1)
We will show
for n 3.
H 2 (NT ; n 2 (1)) = 0
46
T
(with
Consider the loalization sequene
0!Sha2T ( n 2 (1)),!H 2 (NT ; n 2 (1))! vjT H 2 (Nv ; n 2 (1))
that denes the vetor spae Sha2 ( n 2 (1)). By loal duality,
H 2 (Nv ; n 2 (1)) ' H 0 (Nv ; 2 n ) = 0
sine the representation 2 n is potentially unramied or
potentially rystalline.
47
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.
48
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.
49
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.
50
Therefore,
dimHf1 ( ; Wn ) n < dimHf1 ( p ; Wn ) 2n
for n suÆiently large, yielding niteness of
X (Z S )
in any ase.
However, the eetivity in n that appears in (0.4) should eventually
apply to the problem of nding points.
51
VIII. Comments
Non-abelian priniple of Birh and Swinnerton-Dyer:
non-vanishing of (most) L-values ) bounds for 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 full
piture, where, for example, the non-vanishing of L should be a
non-abelian statment.
Also, both impliations should eventually be diret.
52
Setion onjeture and non-abelian desent:
A standard onjeture (say, Bloh-Kato) + setion onjeture ) an
algorithm for nding all points in X (Z S ), in priniple.
53