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 pⁿ-division points on E. Write E_p^∞ 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, ℤ_p), 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 Hⁱ(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

1

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(ℚ_υ) : E₀(ℚ_υ)], where, as usual, E₀(ℚ_υ) 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,

2

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_∞)) = ρ_p(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 ℤ_p. 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

3

we recall that Γ_∞ has p-cohomological dimension equal to 1, so that χ(Γ_∞, A) = #(H⁰(Γ_∞, A))/#(H¹(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

4

where U runs over all open subgroups of H_∞, be the Iwasawa algebra of H_∞. Write  = Hom (A, ℚ_p/ℤ_p) 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

5

where we are now interpreting the elements of I(Γ_∞) as ℤ_p-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

6

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 F₀ = ℚ(E_p), and let Σ_∞ denote the Galois group of F_∞ over F₀, 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_∞/F₀). Then Conjecture 1 holds, and Hⁱ(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 Hⁱ(G_∞, E_p^∞) (i ⩾ 0) are finite.

Theorem 3. Under the same hypotheses as in Conjecture 1, we have that H⁰(G_∞, 𝒮(F_∞)) is finite, and

7

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 ℚ_s 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(𝔽_p))²⋅∏_υc_υ^(p).

Proof. This is a purely local calculation. For each υ ∈ S, fix a place ω of F_∞ above υ, and let Δ_ω denote the Galois group of F_∞,ω over ℚ_υ. 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 ℚ_p is deeply ramified in the sense of ref. 8 because it contains the deeply ramified field ℚ_p(μ_p^∞), where μ_p^∞ denotes the group of all p-power roots of unity. We can therefore apply the principal results of ref. 8 to calculate Ker γ_p and Coker γ_p. We deduce that γ_p is surjective because H²(Δ_ω, Ẽ_p^∞) = 0 and that Ker γ_p 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) H²(G(ℚ_S|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 ℤ_p-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

8

is exact, where H_∞,υ = ⊕_ω H¹(ℚ_∞,ω, E) (p) and ω runs over all places of ℚ_∞ dividing υ. Next, we assert that H¹(Γ_∞, 𝒮(ℚ_∞)) = 0. Indeed, H¹(Γ_∞, 𝒮(ℚ_∞)) is finite because 𝒮(ℚ) is finite, whence H¹(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 H⁴(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 Hⁱ(G_∞, J_∞,υ) = 0 for all i ⩾ 1.

Proof. Fix a place ω of F_∞ above υ, and let Δ_ω denote the Galois group of F_∞,ω over ℚ_υ. Then for all i ⩾ 0, we have

On the other hand, the results of ref. 8 show that H¹(F_∞,ω, E)(p) is isomorphic as a Δ_∞-module to A_ω, where A_ω is defined to be H¹(F_∞,ω, B_ω), with B_ω = E_p^∞ or Ẽ_p^∞, according as υ ≠ p or υ = p. One then proves that Hⁱ(Δ_ω, B_ω) = 0 for all i ⩾ 2. Using the Hochschild-Serre spectral sequence, it is then easy to show that Hⁱ(Δ_ω, 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 pⁿ 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 F₀.

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 ℤ_p-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 GL₂(ℤ_p), one sees that, for all sufficiently large n, the norm map from F_n to F_n−1 acts as multiplication by p⁴ 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

9

and that the sequence

10

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 Hⁱ(G_∞, H¹(G(ℚ_S/F_∞), E_p^∞)) = 0 for i ⩾ 2, and

11

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(ℚ_S/F_∞) has p-cohomological dimension ⩽2, together with the fact that H⁴(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 Hⁱ(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 ρ_p(E/ℚ). If we assume that is torsion over I(Σ_∞), we can prove this conjecture for the Euler characteristic of 𝒮(K_∞).