Iwasawa theory for Symmetric Square of non-$p$-ordinary eigenforms

TL;DR

This paper formulates and provides evidence for Iwasawa main conjectures related to the symmetric square of certain modular forms, utilizing Beilinson--Flach elements and their factorization.

Contribution

It introduces integral signed Iwasawa main conjectures for symmetric square motives and demonstrates their validity through factorization of Beilinson--Flach elements.

Findings

Factorization of Beilinson--Flach elements into integral signed elements

Evidence for the existence of a rank-two Euler system

Proved one inclusion in the Iwasawa main conjectures

Abstract

Let $f$ be a normalized cuspidal eigen-newform of level coprime to $p$ with $a_p(f)=0$. We formulate both integral signed Iwasawa main conjectures and analytic Iwasawa man conjectures attached to the symmetric square motive of $f$ twisted by an auxiliary Dirichlet character. We show that the Beilinson--Flach elements attached to the symmetric square motive factorize into integral signed Beilinson--Flach elements, giving evidence towards the existence a rank-two Euler system predicted by Perrin-Riou. We use these integral elements to prove one inclusion in the integral and analytic Iwasawa main conjectures.