Big Heegner point Kolyvagin system for a family of modular forms

TL;DR

This paper extends Kolyvagin's descent to the big Heegner point Euler system for Hida families, enabling interpolation of Kolyvagin systems across modular forms and advancing Iwasawa theory applications.

Contribution

It develops a method to interpolate and control Tamagawa factors for Hida families, constructing a big Kolyvagin system from individual systems, with applications to Iwasawa theory.

Findings

Constructed a big Kolyvagin system interpolating Fouquet's systems.

Controlled Tamagawa factors at bad primes for Hida families.

Applied results to Iwasawa theory contexts.

Abstract

The principal goal of this paper is to develop Kolyvagin's descent to apply with the big Heegner point Euler system constructed by Howard for the big Galois representation $\mathbb{T}$ attached to a Hida family $\mathbb{F}$ of elliptic modular forms. In order to achieve this, we interpolate and control the Tamagawa factors attached to each member of the family $\mathbb{F}$ at bad primes, which should be of independent interest. Using this, we then work out the Kolyvagin descent on the big Heegner point Euler system so as to obtain a big Kolyvagin system that interpolates the collection of Kolyvagin systems obtained by Fouquet for each member of the family individually. This construction has standard applications to Iwasawa theory, which we record at the end.