EULER CHARACTERISTICS AND ELLIPTIC-CURVES

Let E be a modular elliptic curve over Q, without complex multiplicati on; let p be a prime number where E has good ordinary reduction; and l et F infinity be the field obtained by adjoining to Q all p-power divi sion points on E. Write G infinity for the Galois group of F infinity over Q. Assume that the complex L-series of E over Q does not vanish a t s = 1. If p greater than or equal to 5, we make a precise conjecture about the value of the G infinity-Euler characteristic of the Selmer group of E over F infinity. If one makes a standard conjecture about t he behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations i n the proof depend on recent joint work of the first author with R. Gr eenberg.