Results from MathSciNet: Mathematical Reviews on the Web \begin{matrix}c \\ ◯ \end{matrix} Copyright American Mathematical Society 2000

95a:11059 11G40 11F67 11F85 11G05 11G15

Greenberg, Ralph ^{(}1-WA^{)}

Elliptic curves and p-adic deformations. Elliptic curves and related topics, 101–110, CRM Proc. Lecture Notes, 4, Amer. Math. Soc., Providence, RI, 1994. Let E denote an elliptic curve over Q, S_{E}(Q) its Selmer group, and w(E) the sign of its functional equation. In this very interesting paper, the author discusses the following “proposed theorem”: If w(E) = −1 and p ≥ 5 is a prime at which E has good ordinary reduction, then the p-primary part of S_{E}(Q) is infinite. This would follow from the Birch-Swinnerton-Dyer conjecture. For the case of complex multiplication, the proposed theorem is known to be true, and the author includes a new proof based on a theorem of Rohrlich about special values of L-functions of Hecke characters. For the general case, he shows, using the p-adic Galois representations constructed by Hida, that the proposed theorem follows from the following conjecture: Let Σ be a finite set of primes. Let f run over all normalized eigenforms of weight 2 for Γ_{0}(M), where M is divisible only by primes in Σ. Then, except for finitely many such f’s, the order of vanishing of the L-function L(f, s) at s = 1 is 0 or 1. What is actually needed in the proof is to consider levels of the form M = Np^{ν}, where N is fixed and ν ≥ 0 varies. The two proofs, for the CM case and for the general case, are closely analogous, with Rohrlich’s theorem being the CM analogue of the conjecture above. This close analogy is one of the more interesting aspects of the paper. In the final section, the author shows that a similar argument will also work for forms of higher weight, provided one assumes a generalized form of the conjecture.

{For the entire collection see MR 94i:11002}. Fernando Q. Gouvea (1-COLBY)