Kolyvagin's Work On The Euler System of Heegner Points and Rank 1 BSD

Abstract

We give a new exposition of Kolyvagin's work on the Euler system of Heegner points, which proves the rank 1 (weak) Birch and Swinnerton-Dyer conjecture for modular elliptic curves. Kolyvagin's work also obtains a result towards the finiteness of the Tate-Shafarevich group, proving it has no p-torsion for all but finitely many primes p. We also include some background material on elliptic curves in the appendices. This thesis is the result of a year-long reading project which mainly used Gross's paper summarizing Kolyvagin's results, and Silverman's book for background knowledge.