A Nonvanishing Theorem for Derivatives of Automorphic L-functions with Applications to Elliptic Curves

1. A brief history of nonvanishing theorems. The nonvanishing of a Dirichlet series 2 a(n)n~\ or the existence of a pole, at a particular value of s often has applications to arithmetic. Euler gave the first example of this, showing that the infinitude of the primes follows from the pole of Ç(s) at s = 1. A deep refinement was given by Dirichlet, whose theorem on primes in an arithmetic progression depends in a fundamental way upon the nonvanishing of Dirichlet L-functions at s = 1. Among the many examples of arithmetically significant nonvanishing results in this century, one of the most important is still mostly conjectural. Let E be an elliptic curve defined over Q: the set of all solutions to an equationy = x-ax-b where a, b are rational numbers with 4a-27b ^ 0. Mordell showed that E(Q) may be given the structure of a finitely generated abelian group. The Birch-Swinnerton-Dyer Conjecture asserts that the rank of this group is equal to the order of vanishing of a certain Dirichlet series L(s,E) as s = 1—the center of the critical strip—and that the leading Taylor coefficient of this L-function at s = 1 is determined in an explicit way by the arithmetic of the elliptic curve. We refer to the excellent survey article of Goldfeld [5] for details. In 1977, Coates and Wiles [3] proved the first result towards the BirchSwinnerton-Dyer conjecture. The conjecture implies that if the L-series of E does not vanish at 1, then the group of rational points is finite. Coates and Wiles proved this last claim in the special case that E has complex multiplication (nontrivial endomorphisms). In this note, we announce a nonvanishing theorem which, together with work of Kolyvagin and Gross-Zagier, implies that E(Q) is finite when L(l,E) ^ 0 for any modular elliptic curve E. (A modular elliptic curve is one which may be parametrized by automorphic functions. Deuring proved that all elliptic curves with complex multiplication are modular; Taniyama and Weil have conjectured that indeed all elliptic curves defined over Q are modular.) Before giving details of our theorem, we mention several other nonvanishing theorems and arithmetic applications. The following discussion is necessarily not a complete survey Shimura showed that there is a correspondence between modular forms ƒ of even weight k and modular forms ƒ of half-integral weight (k + l)/2.