Selmer groups and the Eisenstein-Klingen ideal

TL;DR

This paper develops a strategy to establish a divisibility relation between the p-adic L-function and the Selmer group characteristic ideal for twisted Galois representations associated with Hida families, advancing the Iwasawa main conjecture.

Contribution

It introduces a third characteristic ideal involving congruences between Siegel modular forms and Eisenstein series, proving one divisibility in the Iwasawa main conjecture context.

Findings

Proved the divisibility of the Eisenstein ideal by the Selmer group characteristic ideal.

Established a framework connecting congruences of modular forms to Iwasawa theory.

Contributed to the proof of a key divisibility in the main conjecture for Galois representations.

Abstract

In this article, we set up a strategy to prove one divisibility towards the main Iwasawa conjecture for the Selmer groups attached to the twisted adjoint modular Galois representations associated to Hida families. This conjecture asserts the equality of the p-adic L-function interpoling the critical values of the symmetric square of the modular forms in these families and the characteristic ideal of the associated Selmer group. The idea is to introduce a third characteristic ideal containing informations on the congruences between cuspidal Siegel modular forms of genus 2 and the Klingen type Eisenstein series and to prove the two divisibilities: The p-adic L-function divides the Eisenstein ideal and that the Eisenstein ideal divides the characteristic ideal of the Selmer group. In that paper we proved the latter divisibility.