Euler systems and modular elliptic curves

Abstract

This paper consists of two parts. In the first we present a general theory of Euler systems. The main results (see §§3 and 4) show that an Euler system for a p-adic representation T gives a bound on the Selmer group associated to the dual module Hom(T, μp∞). These theorems, which generalize work of Kolyvagin [Ko], have been obtained independently by Kato [Ka1], Perrin-Riou [PR2], and the author [Ru3]. We will not prove these theorems here, or even attempt to state them in the greatest possible generality. In the second part of the paper we show how to apply the results of Part I and an Euler system recently constructed by Kato [Ka2] (see the article of Scholl [Scho] in this volume) to obtain Kato’s theorem in the direction of the Birch and Swinnerton-Dyer conjecture for modular elliptic curves (Theorem 8.1).