Quaternionic Kolyvagin systems and Iwasawa theory for Hida families

TL;DR

This paper constructs a modified universal Kolyvagin system for Galois representations of Hida families, extending previous work to quaternionic settings and providing evidence for the anticyclotomic Iwasawa main conjecture.

Contribution

It generalizes Büyükkoduk's work to quaternionic settings and relaxes the Heegner hypothesis for Hida families.

Findings

Constructed a modified universal Kolyvagin system for Hida families.

Proved one divisibility of the anticyclotomic Iwasawa main conjecture.

Abstract

We build a modified universal Kolyvagin system for the Galois representation attached to a Hida family of modular forms, starting from the big Heegner point Euler system of Longo--Vigni built in towers of Shimura curves. We generalize the work of B\"uy\"ukboduk to a quaternionic setting, relaxing the classical \emph{Heegner hypothesis} on the tame conductor of the family. As a byproduct of this construction, we give a proof of one divisibility of the anticyclotomic Iwasawa main conjecture for Hida families.