Description:

Title: Kolyvagin's Conjecture and Higher Congruences of Modular Forms.

Abstract: 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 deduced remarkable consequences for the Selmer rank of E. For example, his results, combined with work of Gross-Zagier, implied that a curve with analytic rank one also has algebraic rank one; a partial converse follows from his conjecture. In this talk, I will report on work that proved several new cases of Kolyvagin's conjecture. The methods follow the groundbreaking work of Wei Zhang, who used congruences between modular forms to prove Kolyvagin's conjecture under some technical hypotheses. By considering congruences modulo higher powers of p, we remove many of those hypotheses. The talk will provide a brief review of Kolyvagin's conjecture and its applications, explain an analog of the conjecture in an opposite parity regime, and give an overview of the proofs, including the difficulties associated with higher congruences of modular forms and how they can be overcome via deformation theory.