Iwasawa theory for Rankin--Selberg products of $p$-non-ordinary eigenforms

TL;DR

This paper advances Iwasawa theory for non-ordinary modular forms by linking Euler systems, $p$-adic $L$-functions, and cohomology, confirming conjectures and establishing growth formulas in the cyclotomic tower.

Contribution

It proves that four Euler systems for $f imes g$ originate from a single cohomology class, and introduces an explicit Wach module-based logarithmic matrix to describe their growth.

Findings

Confirmed a conjecture relating Euler systems and cohomology classes.

Derived an explicit growth formula for Euler systems and $p$-adic $L$-functions.

Proved one inclusion of the signed Iwasawa main conjecture under hypotheses.

Abstract

Let $f$ and $g$ be two modular forms which are non-ordinary at $p$. The theory of Beilinson-Flach elements gives rise to four rank-one non-integral Euler systems for the Rankin-Selberg convolution $f \otimes g$, one for each choice of $p$-stabilisations of $f$ and $g$. We prove (modulo a hypothesis on non-vanishing of $p$-adic $L$-fuctions) that the $p$-parts of these four objects arise as the images under appropriate projection maps of a single class in the wedge square of Iwasawa cohomology, confirming a conjecture of Lei-Loeffler-Zerbes. Furthermore, we define an explicit logarithmic matrix using the theory of Wach modules, and show that this describes the growth of the Euler systems and $p$-adic $L$-functions associated to $f \otimes g$ in the cyclotomic tower. This allows us to formulate "signed" Iwasawa main conjectures for $f\otimes g$ in the spirit of Kobayashi's $\pm$-Iwasawa theory for supersingular elliptic curves; and we prove one inclusion in these conjectures under our running hypotheses.