Kolyvagin’s conjecture, bipartite Euler systems, and higher congruences of modular forms

SEMINARS, NUMBER THEORY

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February 16, 2022 3:00 pm - 4:00 pm
Science Center 507
Address: 1 Oxford Street, Cambridge, MA 02138 USA
Speaker:

Naomi Sweeting - Harvard


For an elliptic curve E,  Kolyvagin used Heegner points to construct special Galois cohomology classes valued in the torsion points of E. Under the conjecture that not all of these classes vanish, he showed that they encode the Selmer rank of E. I will explain a proof of new cases of this conjecture that builds on prior work of Wei Zhang. The proof naturally leads to a generalization of Kolyvagin's work in a complimentary "definite" setting, where Heegner points are replaced by special values of a quaternionic modular form. I'll also explain an "ultrapatching" formalism which simplifies the Selmer group arguments required for the proof.