On the vanishing of almost all primary components of the Shafarevich-Tate group of elliptic curves over the rationals

Abstract

The Shafarevich-Tate and Selmer groups arise in the context of Kummer theory for elliptic curves. The finiteness of the Shafarevich-Tate group of an elliptic curve $E$ over the field of rational numbers is included in the Birch and Swinnerton-Dyer conjectures, and is still an open question. We present an overview of the Shafarevich-Tate and Selmer groups of an elliptic curve in the framework of Galois cohomology. Known results on the finiteness of the Shafarevich-Tate group are mentioned, including results of Coates and Wiles, Rubin, Gross and Zagier, and Kolyvagin. We then prove the vanishing of the $\ell$-primary component of the Shafarevich-Tate group for almost all primes $\ell$, for any elliptic curve $E$ over the rationals without complex multiplication.