Euler characteristics and elliptic curves
Abstract
Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F ∞ be the field obtained by adjoining to ℚ all p-power division points on E. Write G∞ for the Galois group of F ∞ over ℚ. Assume that the complex L-series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G ∞-Euler characteristic of the Selmer group of E over F ∞. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.
Let E be an elliptic curve defined over ℚ. For
simplicity, we shall assume throughout that E does not admit
complex multiplication. Let p be a prime number, and write
E
pn (n = 1, 2, …) for
the group of p
n-division points on E.
Write Ep∞ for the union of the
E
pn (n = 1, 2, …). Put
F
∞ = ℚ(E
p∞), and let
G
∞ denote the Galois group of
F
∞ over ℚ. By a theorem of Serre (1),
G
∞ is an open subgroup of GL(2,
ℤp), and hence is a p-adic Lie group of
dimension 4. Assume from now on that p ⩾ 5, so that
G
∞ has no p-torsion. By a
refinement (2) of a theorem of Lazard (3), G
∞
then has p-cohomological dimension equal to 4. Let
A be a p-primary abelian group, which is a
discrete G
∞-module. We say that A
has a finite G
∞-Euler characteristic if all of
the cohomology groups
H
i(G
∞, A) (i ⩾
0) are finite. When A has finite
G
∞-Euler characteristic, we define its Euler
characteristic χ(G
∞, A) by
The present note will be concerned with the calculation of the
G
∞-Euler characteristic of the Selmer group
𝒮(F
∞) of E over
F
∞. We recall that this Selmer group is
defined by the exactness of the sequence
where ω runs over all finite places of
F
∞; here F
∞,ω
denotes the union of the completions at ω of the finite extensions of
ℚ contained in F
∞. Of course,
𝒮(F
∞) has a natural structure as a
G∞-module, and we expect its Euler characteristic to be
closely related to the Birch and Swinnerton-Dyer formula. Specifically,
let III (E) denote the Tate-Shafarevich group of E over ℚ,
and, for each finite prime υ, let c
υ =
[E(ℚυ) :
E
0(ℚυ)], where, as usual,
E0(ℚυ) is the subgroup of points with
nonsingular reduction modulo υ. Let L(E, s) be the
Hasse-Weil L-series of E over ℚ. If
B is an abelian group, we write B(p) for its
p-primary subgroup. If n is a positive integer,
n
(p)will denote the exact power of
p dividing n. Finally, we denote by
Ẽ the reduction of E modulo p.
We then define, for p where E has good reduction,
where υ runs over all finite places of ℚ.
Conjecture 1. Let E be a modular elliptic curve over ℚ, without complex multiplication, such that L(E, 1) ≠ 0. Let p ⩾ 5 be a prime number such that E has good ordinary reduction at p. Then 𝒮(F ∞) has a finite G∞-Euler characteristic, which is given by χ(G ∞, 𝒮(F ∞)) = ρp(E/ℚ).
This conjecture is suggested by the following considerations in
Iwasawa theory. Let ℚ∞ denote the unique extension of
ℚ such that the Galois group Γ∞ of ℚ∞
over ℚ is isomorphic to ℤp. Of course,
ℚ∞ is contained in F
∞. Let
𝒮(ℚ∞) be the Selmer group of E over
ℚ∞, which is defined by replacing F∞ by
ℚ∞ in the exact sequence of Eq. 1. Making
the same hypotheses on E and p as in Conjecture
1, it is well known that 𝒮(ℚ∞) has a finite
Γ∞-Euler characteristic, which is given by
we recall that Γ∞ has p-cohomological
dimension equal to 1, so that χ(Γ∞, A) =
#(H
0(Γ∞,
A))/#(H
1(Γ∞,
A)) for any discrete p-primary
Γ∞-module A. Thus Conjecture 1 asserts that,
under the hypotheses made on E and p, the
G∞-Euler characteristic of
𝒮(F
∞) should be precisely equal to the
Γ∞-Euler characteristic of 𝒮(ℚ∞).
This is indeed what one would expect from the following heuristic
argument. If H
∞ is any profinite group, let
where U runs over all open subgroups of
H
∞, be the Iwasawa algebra of
H
∞. Write  = Hom
(A,
ℚp/ℤp) for the
Pontrjagin dual of a discrete p-primary abelian group
A. Under the hypotheses of Conjecture 1, it is
known that 
is a
finitely generated torsion module over
I(Γ∞), whereas the structure theory of such
modules enables us to define the characteristic ideal
C(𝒮(ℚ∞)) of 𝒮̂(ℚ∞)
in I(Γ∞). It is easy and well known to see
that C(𝒮(ℚ∞)) has a generator
μ(ℚ∞) such that
where we are now interpreting the elements of
I(Γ∞) as ℤp-valued
measures on Γ∞. We do not at present know enough about
the structure theory of I(G
∞)-modules to be
able to define the analogue C(𝒮(F
∞)) of
C(𝒮(ℚ∞)). Nevertheless, one is
tempted to guess that there should be a generator
μ(F
∞) of C(𝒮(F
∞))
such that
Moreover, the link, which may exist between these characteristic
ideals and p-adic L-functions, suggests that
C(𝒮(F
∞)) should map to
C(𝒮(ℚ∞)) under the canonical surjection
from I(G
∞) onto
I(Γ∞). This latter property would show that
the two integrals on the left of Eqs. 4 and 5 are
equal, for suitable generators of C(𝒮(F
∞))
and C(𝒮(ℚ∞)), and so explain the equality
of the Euler characteristics.
In spite of the above heuristic argument, it does not seem easy to prove Conjecture 1. Let F 0 = ℚ(E p), and let Σ∞ denote the Galois group of F ∞ over F 0, so that Σ∞ is a pro-p-group. We say that a module X over the Iwasawa algebra I(Σ∞) is torsion if each element of X is annihilated by some non-zero element of I(Σ∞). Our main result is the following.
Theorem 2. In addition to the hypotheses of
Conjecture 1, assume that
is torsion
over the Iwasawa algebra I(Σ∞), where
Σ∞ = G(F
∞/F
0). Then Conjecture 1 holds, and
H
i(G
∞,
𝒮(F
∞)) = 0 for i = 2, ⋯ , 4.
It has long been conjectured (see ref. 4) that
is torsion over
I(Σ∞) for all E and all primes
p where E has good ordinary reduction, but very
little is known in this direction at present. In view of this, it may
be worth noting the following weaker result, which we can prove without
this assumption. By a theorem of Serre (5), the cohomology groups
H
i(G
∞, E
p∞) (i
⩾ 0) are finite.
Theorem 3. Under the same hypotheses as in
Conjecture 1, we have that
H
0(G
∞,
𝒮(F
∞)) is finite, and
Sketch of the Proof of Theorem 3.
Let S be a fixed
finite set of nonarchimedean primes containing p and all
primes where E has bad reduction. We write
ℚs for the maximal extension of ℚ unramified
outside S and ∞. For each n ⩾ 0, let
F
n = ℚ(E
pn+1). We
define, for υ ∈ S,
where ω runs over all primes of F
n
dividing υ, and the inductive limit is taken with respect to the
restriction maps. Our proof is based on the following well known
commutative diagram with exact rows
where the vertical arrows are restriction maps.
Lemma 4. The map γ is surjective, and its kernel is finite of order #(Ẽ(𝔽p))2⋅∏υ c υ (p).
Proof. This is a purely local calculation. For each
υ ∈ S, fix a place ω of F
∞
above υ, and let Δω denote the Galois
group of F
∞,ω over
ℚυ. Assume first that υ ≠ p. Then
and simple calculations (cf. ref. 7, Lemma 13) then show that
Suppose next that υ = p. The extension
F
∞,ω of ℚp is
deeply ramified in the sense of ref. 8 because it contains the deeply
ramified field
ℚp(μp∞), where
μp∞ denotes the group of all
p-power roots of unity. We can therefore apply the principal
results of ref. 8 to calculate Ker γp and
Coker γp. We deduce that
γp is surjective because
H
2(Δω,
Ẽp∞) = 0 and that Ker
γp is finite, with order equal to
completing the proof of the lemma.
Lemma 5. Assume L(E, 1) ≠ 0. Then (i) 𝒮(ℚ) is finite, (ii) H 2(G(ℚS|ℚ), E p∞) = 0, and (iii) the cokernel of λ is finite of order equal to #(E(ℚ)(p)).
Proof. Assertion (i) is a fundamental result of Kolyvagin. Assertions (ii) and (iii) follow immediately from the finiteness of 𝒮(ℚ) and Cassels’ variant of the Poitou-Tate sequence (cf. the proof of Theorem 12 of ref. 7).
Lemma 6. Assume that L(E, 1) ≠ 0. Then the map λ∞ in the above diagram is surjective.
Proof. We make essential use of the cyclotomic
ℤp-extension ℚ∞ of
ℚ. The finiteness of 𝒮(ℚ) implies that
is torsion over
the Iwasawa algebra I(Γ∞). A well known
argument then shows that the sequence
is exact, where H
∞,υ = ⊕ω
H
1(ℚ∞,ω, E) (p) and ω runs over
all places of ℚ∞ dividing υ. Next, we assert that
H
1(Γ∞, 𝒮(ℚ∞)) =
0. Indeed, H
1(Γ∞,
𝒮(ℚ∞)) is finite because 𝒮(ℚ) is finite,
whence H
1(Γ∞,
𝒮(ℚ∞)) = 0 because
has no non-zero
finite Γ∞-submodule (see ref. 9). Hence, taking
Γ∞-invariants of the above exact sequence, we see that
the natural map
is surjective. But the surjectivity of ϕ∞ and the
surjectivity of γ together clearly show that γ∞ is
surjective, as required.
Lemma 7 (J.-P. Serre, personal communication). We have χ(G ∞, E p∞) = 1 and H 4(G ∞, E p∞) = 0.
To prove Theorem 3, one simply uses diagram chasing in the above diagram, combined with Lemmas 4–7.
Sketch of the Proof of Theorem 2.
We begin with another purely local calculation. For each υ ∈ S, let J ∞,υ be the G ∞-module defined at the beginning of §2.
Lemma 8. For each υ ∈ S, we have H i(G ∞, J ∞,υ) = 0 for all i ⩾ 1.
Proof. Fix a place ω of F
∞
above υ, and let Δω denote the Galois group of
F
∞,ω over ℚυ. Then for all
i ⩾ 0, we have
On the other hand, the results of ref. 8 show that
H
1(F
∞,ω, E)(p) is isomorphic as
a Δ∞-module to A
ω, where
A
ω is defined to be
H
1(F
∞,ω, B
ω), with
B
ω = E
p∞ or
Ẽp∞, according as υ ≠
p or υ = p. One then proves that
H
i(Δω, B
ω) =
0 for all i ⩾ 2. Using the Hochschild-Serre
spectral sequence, it is then easy to show that
H
i(Δω, A
ω) =
0 for all i ⩾ 1, as required.
If W is an abelian group, we define, as usual,
T
p(W) =
lim← (W)pn,
where (W)pn denotes the kernel of
multiplication by p
n on W. We put
T
p(E) for
T
p(E
p∞). For each integer
m ⩾ 0, we define R(F
m) by the
exactness of the sequence
We then define
where the projective limit is taken with respect to the
corestriction maps from F
m to
F
n when m ⩾ n. Recall that
Σ∞ denotes the Galois group of
F
∞ over F
0.
Lemma 9. If
is torsion
over the Iwasawa algebra
I(Σ∞), then
ℛ(F
∞) = 0.
Proof. This is analogous to the well known argument for the
cyclotomic ℤp-extension ℚ∞,
which has already been implicitly used in proving exactness at the
right hand end of Eq. 8 (we recall that
L(E, 1) ≠ 0 automatically implies that
is torsion over
I(Γ∞)). The only unexpected point is to note
that the projective limit of the E
pn+1(n =
0, 1, …) with respect to the norm maps from
F
m to F
n when
m ⩾ n is in fact zero. Indeed, since
G
∞ is open in
GL
2(ℤp), one sees that, for all
sufficiently large n, the norm map from
F
n to F
n−1 acts as
multiplication by p
4 onto
E
pn+1, whence the previous assertion is
plain.
We assume that for the rest of this section that
𝒮(F
∞) is torsion over the Iwasawa algebra
I(Σ∞). Then we claim that
and that the sequence
is exact. Indeed, applying Cassels’ variant of the Poitou-Tate
sequence to each of the fields F
n(n = 0,
1, …), and then passing to the inductive limit as n
→ ∞ with respect to the restriction maps, we obtain an exact
sequence
whence Eqs. 9 and 10 follow immediately from
Lemma 9. In fact, Eq. 9 is known to be true for all
p ≠ 2 without any additional hypothesis.
Lemma 10. Assume that
is torsion over
I(Σ∞). Then
H
i(G
∞,
H
1(G(ℚS/F
∞),
E
p∞)) = 0 for
i ⩾ 2, and
Proof. The assertion (Eq. 11)
follows from the Hochschild-Serre spectral sequence (cf. Theorem 3 of
ref. 10) on using Eq. 9 and (ii) of Lemma 5.
Similarly, the first assertion of Lemma 10 is an immediate consequence
of Theorem 3 of ref. 10 and the fact that
G(ℚS
/F∞) has
p-cohomological dimension ⩽2, together with the fact that
H
4(G
∞, E
p∞) = 0
(J.-P. Serre, personal communication).
To complete the proof of Theorem 2, we take
G
∞-invariants of the exact sequence (Eq.
10). Using Lemmas 6, 8, and 10, we deduce that
and that H
i(G
∞,
𝒮(F
∞)) = 0 for all i ⩾ 2. Hence,
Theorem 2 follows from Theorem 3.
We finish with the following remark. Let K
∞ be
the fixed field of the center of G
∞, and let
H
∞ denote the Galois group of
K
∞ over ℚ. We conjecture that, under the
same hypotheses as Conjecture 1, the H
∞-Euler
characteristic of the Selmer group 𝒮(K
∞) of
E over K
∞ is finite and equal to
ρp(E/ℚ). If we assume that
is torsion over
I(Σ∞), we can prove this conjecture for the
Euler characteristic of 𝒮(K
∞).
Acknowledgments
We are very grateful to J.-P. Serre for providing us with a proof that χ(G ∞, E p∞) = 1. We also warmly thank B. Totaro for pointing out to us a result that revealed an error in an earlier version of this manuscript.
Footnotes
- Copyright © 1997, The National Academy of Sciences of the USA
