The priniple of Birh and Swinnerton-Dyer for
ertain hyperboli urves
Minhyong Kim
Otober 23, 2007
1
Priniple of Birh and Swinnerton-Dyer for ellipti urves E=Q :
L(E; 1) 6= 0 ) E (Z ) is nite.
Does this extend (in some sense) to hyperboli urves?
2
Context:
S : a nite set of primes inluding arhimedean one.
X=Q hyperboli urve with good redution outside S . That is, X is
the generi ber of a smooth projetive Z S -urve with an etale
divisor (possibly empty) removed, and X (C ) is a onneted
hyperboli Riemann surfae.
Fix a point b 2 X (Z S ).
3
Need to study map
X (Z S )!H 1 (; 1 (X; b))
x 7! [1 (X ; b; x)℄
in various ontexts.
Grothendiek's letter to Faltings disusses ase of
b):
^1 (X;
Conjetured this map to be bijetive (in the ompat ase), and
onneted to a proof of niteness.
4
Can also take 1 to be the motivi fundamental group, onsisting of
ompatible systems of pro-unipotent ompletions. Arrive at a
fundamental diagram:
X (Z S ) - X (Z p )
un
un;lo
? et
-
? et
d u
r;n
-
-
lop 1
D DR 0
1
Hf ( ; Un )
Hf ( p ; Un )
Un =F
5
The
Hf1 (; Unet )
are loal and global Selmer Varieties. The subsript `f ' refers to
loal onditions imposing that torsors be unramied outside
T = S [ fpg and rystalline at p.
Whenever
dimHf1 ( ; Unet ) < dimHf1 ( p ; Unet )
get niteness for X (Z S ). (Inequality is implied by standard motivi
onjetures.)
6
But should eventually use non-abelian fundamental groups to nd
points, via a non-abelian method of Chabauty and non-abelian
desent.
Non-abelian BSD should fall into this irle of ideas. (Note that
Grothendiek's onjeture is an extension of the niteness
onjeture for Tate-Shafarevih groups.)
7
Consider
X := E n f0g;
where E=Q is an ellipti urve with omplex multipliation by an
imaginary quadrati eld K .
Coates and Wiles resolved (part of) BSD for E using the `method
of p-adi L-funtions'.
Can use parallel ideas to give somewhat expliit niteness for
X (Z S ).
8
Notation:
:= Gal(Q =Q )
N := Gal(Q =K )
p = a prime of good redution for E , split in K
p = Gal(Q p =Q p )
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.
9
Have orresponding p-adi L-funtions:
Lp 2 ; Lp 2 10
p-adi polylogrithms for E :
For dierentials ; of seond kind for X , dene
Pn(z) =
Pn(z) =
Zz
b
Zz
b
n n
These are loally analyti Coleman funtions on X (Z p ).
11
Corollary 0.1
There is a non-trivial polynomial
f = f (P ; P )
of the Pn ; Pn restriting to a non-zero onvergent power series on
eah residue disk of X (Z p ), suh that
f (z ) = 0
for eah point z 2 X (Z S ) X (Z p ).
12
r = dimHf1 ( ; Vp (E ))
s = jS j
Suppose k (Lp ) 6= 0 and k (Lp ) 6= 0 for all
k > 0. Then there is a non-trivial polynomial
Corollary 0.2
f = f (P ; P )
of the Pn ; Pn , for n r + s, restriting to a non-zero onvergent
power series on eah residue disk of X (Z p ), suh that
f (z ) = 0
for eah point z 2 X (Z S ) X (Z p ).
13
Remarks:
-The non-vanishing hypothesis is a folklore onjeture.
-The polynomial f seems in priniple omputable.
14
b).
U : Q p -pro-unipotent etale fundamental group for (X;
U 1 = U , U n = [U; U n 1℄.
Un = U=U n+1 .
Fundamental diagram:
X (Z S )
#
Hf1 ( ; Un )
,!
X (Z p )
#
! Hf1( p ; Un)
x 7! [1Q p (X ; b; x)℄
15
U is somewhat ompliated. Replae with a quotient
-- W
U
with the property that
U2 ' W2
and
W n =W n+1 ' n 2 (1) n 2 (1)
viewed as a representation of in the natural way.
16
Constrution:
= 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
T (U1 ) = T (Vp )
where T ( ) refers to the tensor algebra (but with a dierent
Galois ation).
17
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.
18
However, easy to hek:
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.
19
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)
20
Fundamental diagram an thus be extended to
X (Z S )
#
,!
X (Z p )
#
! Hf1( p; Un)
#
#
Hf1 ( ; Wn ) ! Hf1 ( p ; Wn )
Hf1 ( ; Un )
21
The map
jn : X (Z p )!Hf1 ( p ; Wn )
is desribed by non-abelian p-adi Hodge theory:
Hf1 ( p ; Wn ) ' F 0 nWnDR
aording to whih
jn (Coordinate ring of Hf1 ( p ; Wn ))
is ontained in the ring generated by Pm ; Pm for m n.
22
Meanwhile:
Theorem 0.3
dimHf1 ( ; Wn ) < dimHf1 ( p ; Wn )
for n >> 0.
23
Also:
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.
The earlier orollaries follow immediately from the theorems.
24
Proof of theorem 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.
25
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 (reall T = S [ fpg)
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.
26
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
27
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
28
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 Q ; 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.
29
In partiular, A Q 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.
30
Thus,
Hf1 ( p ; Wn ) = 2n 2 > r + s + n 3 = dimHf1 ( ; Wn )
as soon as n r + s.
31
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. Therefore,
dimHf1 ( ; Wn ) < dimHf1 ( p ; Wn )
for n suÆiently large, yielding niteness of
X (Z S )
in any ase.
32
Key point is the non-abelian method of Coates and Wiles whereby
non-vanishing of L-values leads to bounds for global Selmer
varieties.
33