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

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.

arXiv commentThis version has been accepted for publication in Documenta Mathematica

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Cited by in corpus (3)

• Iwasawa theory for Rankin--Selberg products of $p$-non-ordinary eigenforms
• A higher rank Euler system for the multiplicative group over a totally real field
• Rank--two Euler systems for symmetric squares