The method of Coates and Wiles for integral
points
Minhyong Kim
Otober 12, 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 to hyperboli urves?
2
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 (this part of) BSD for E using the
`method of p-adi L-funtions'.
3
Notation:
S , a set of primes inluding 1 and those of bad redution for E
:= Gal(Q =Q )
N := Gal(Q =K )
p = a prime of good redution for E , split in K
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.
4
Have orresponding p-adi L-funtions:
Lp 2 ; Lp 2 5
p-adi polylogrithms for E :
Choose dierentials ; of seond kind for X and dene, for n 2,
Pn(z) =
Pn(z) =
Zz
b
Zz
b
n n
These are loally analyti Coleman funtions on X (Z p ).
6
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 ).
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r = dimHf1 ( ; Vp (E ))
s = jsj
Corollary 0.2 Suppose
k > 0.
k(
Lp ) 6= 0
and
k (Lp ) 6= 0 for all
Then there is a non-trivial polynomial
f = f (P ; P )
n r + s, restriting to a non-zero onvergent
power series on eah residue disk of X (Z p ), suh that
of the
Pn ; Pn
, for
f (z ) = 0
for eah point
z 2 X (Z S ) X (Z p ).
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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)℄
9
The subsript `f ' orresponds to the ondition that the ohomology
lasses be unramied outside T = S [ fpg and rystalline at p.
10
U is somewhat ompliated. Replae by 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.
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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).
12
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.
13
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.
14
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)
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Fundamental diagram an thus be extended to
X (Z S )
#
,!
X (Z p )
#
! Hf1( p; Un)
#
#
Hf1 ( ; Wn ) ! Hf1 ( p ; Wn )
Hf1 ( ; Un )
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The map
jn : X (Z p )!Hf1 ( p ; Wn )
is desribed by non-abelian p-adi Hodge theory:
Hf1 ( p ; Wn ) ' F 0 nWn
aording to whih
jn (Coordinate ring of Hf1 ( p ; Wn ))
is ontained in the ring generated by Pm ; Pm for m n.
17
Meanwhile:
Theorem 0.3
dim
for
Hf1 ( ; Wn ) < dimHf1 ( p ; Wn )
n >> 0.
18
Also:
Theorem 0.4 Assume
(*)
k(
Lp) 6= 0 and k (Lp) 6= 0 for all k > 0.
Then
dim
for
Hf1 ( ; Wn ) < dimHf1 ( p ; Wn )
n = r + s.
The earlier orollaries follow immediately from the theorems.
19
Proof of theorem uses main onjeture for K . To x ideas, 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,
dimW1 =F 0 = 1
and
for n 2, so that
F 0 [W n =W n+1 ℄ = 0
dimHf1 ( p ; Wn ) = 2 + 2(n 2) = 2n 2
for n 2.
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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 (where 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.
21
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
22
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
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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.
24
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.
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Thus,
Hf1 ( p ; Wn ) = 2n 2 > r + s + n 3 = dimHf1 ( ; Wn )
as soon as n r + s.
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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.
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