Joint IAS/Princeton University Number Theory Seminar

Given an elliptic curve $E$, Kolyvagin used CM points on modular curves to construct a system of classes valued in the Galois cohomology of the torsion points of $E$. Under the conjecture that not all of these classes vanish, he gave a description for the Selmer group of $E$. This talk will report on recent work proving new cases of Kolyvagin's conjecture. The proof builds on work of Wei Zhang, who used congruences between modular forms to prove Kolyvagin's conjecture under some technical hypotheses. We remove many of these hypotheses by considering congruences modulo higher powers of $p$. The talk will explain the difficulties associated with higher congruences of modular forms and how they can be overcome.