ellipticcurves
Copyright © 1997, The National Academy of Sciences of the USA Colloquium Paper Euler characteristics andellipticcurves | |||||||||||||||||||||||||||||||||||||||||||||||||
Abstract Let E be a modular elliptic curve over , 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
all p-power division points on E. Write
G∞ for the Galois group of F∞
over . Assume that the complex L-series of E
over 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 . For
simplicity, we shall assume throughout that E does not admit
complex multiplication. Let p be a prime number, and write
Epn (n = 1, 2,…) for
the group of pn-division points on E.
Write Ep∞ for the union of the
Epn (n = 1, 2,…). Put
F∞ = (Ep∞), and let
G∞ denote the Galois group of
F∞ over . By a theorem of Serre (1),
G∞ is an open subgroup of GL(2,
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
Conjecture1. Let E be a modular elliptic curve over , 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 (F∞) has a finite G∞-Euler characteristic, which is given by χ(G∞, (F∞)) = ρp(E/). This conjecture is suggested by the following considerations in Iwasawa theory. Let ∞ denote the unique extension of such that the Galois group Γ∞ of ∞ over is isomorphic to p. Of course, ∞ is contained in F∞. Let (∞) be the Selmer group of E over ∞, which is defined by replacing F∞ by ∞ in the exact sequence of Eq. 1. Making the same hypotheses on E and p as in Conjecture 1, it is well known that (∞) has a finite Γ∞-Euler characteristic, which is given by
In spite of the above heuristic argument, it does not seem easy to prove Conjecture 1. Let F0 = (Ep), 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. Theorem2. 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∞, (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∞, Ep∞) (i 0) are finite. Theorem3. Under the same hypotheses as in Conjecture 1, we have that H0(G∞, (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 s for the maximal extension of unramified outside S and ∞. For each n 0, let Fn = (Epn+1). We define, for υ S, Lemma4. The map γ is surjective, and its kernel is finite of order #(Ẽ(p))2υcυ(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 υ. Assume first that υ ≠ p. Then Lemma5. Assume L(E, 1) ≠ 0. Then (i) () is finite, (ii) H2(G(S|), Ep∞) = 0, and (iii) the cokernel of λ is finite of order equal to #(E()(p)). Proof. Assertion (i) is a fundamental result of Kolyvagin. Assertions (ii) and (iii) follow immediately from the finiteness of () and Cassels’ variant of the Poitou-Tate sequence (cf. the proof of Theorem 12 of ref. 7). Lemma6. Assume that L(E, 1) ≠ 0. Then the map λ∞ in the above diagram is surjective. Proof. We make essential use of the cyclotomic p-extension ∞ of . The finiteness of () implies that is torsion over the Iwasawa algebra I(Γ∞). A well known argument then shows that the sequence
Lemma7 (J.-P. Serre, personal communication). We have χ(G∞, Ep∞) = 1 and H4(G∞, Ep∞) = 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. Lemma8. 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 Δω denote the Galois group of F∞,ω over υ. Then for all i 0, we have If W is an abelian group, we define, as usual, Tp(W) = lim← (W)pn, where (W)pn denotes the kernel of multiplication by pn on W. We put Tp(E) for Tp(Ep∞). For each integer m 0, we define R(Fm) by the exactness of the sequence Lemma9. If is torsion over the Iwasawa algebra I(Σ∞), then (F∞) = 0. Proof. This is analogous to the well known argument for the cyclotomic p-extension ∞, 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 Epn+1(n = 0, 1,…) with respect to the norm maps from Fm to Fn when m n is in fact zero. Indeed, since G∞ is open in GL2(p), one sees that, for all sufficiently large n, the norm map from Fn to Fn−1 acts as multiplication by p4 onto Epn+1, whence the previous assertion is plain. We assume that for the rest of this section that (F∞) is torsion over the Iwasawa algebra I(Σ∞). Then we claim that
Lemma10. Assume that is torsion over I(Σ∞). Then Hi(G∞, H1(G(S/F∞), Ep∞)) = 0 for i 2, and
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 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 . We conjecture that, under the same hypotheses as Conjecture 1, the H∞-Euler characteristic of the Selmer group (K∞) of E over K∞ is finite and equal to ρp(E/). If we assume that is torsion over I(Σ∞), we can prove this conjecture for the Euler characteristic of (K∞). | |||||||||||||||||||||||||||||||||||||||||||||||||
Acknowledgments We are very grateful to J.-P. Serre for providing us with a proof that χ(G∞, Ep∞) = 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. | |||||||||||||||||||||||||||||||||||||||||||||||||
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