Trito-non-ordinary Iwasawa theory of diagonal cycles

TL;DR

This paper develops a new Iwasawa theory framework for diagonal cycles associated with specific automorphic forms, introducing signed Euler systems and extending p-adic L-function constructions to a novel non-ordinary setting.

Contribution

It introduces and studies the Euler system of signed diagonal cycles for a trito-non-ordinary triple product, extending Iwasawa theory and p-adic L-functions to this new context.

Findings

Formulated a signed Iwasawa main conjecture with one proven inclusion.

Extended Hsieh's balanced triple-product p-adic L-function to non-ordinary cases.

Developed signed anticyclotomic Iwasawa theory for certain base changes.

Abstract

Our goal in this paper is to introduce and study the Euler system of signed diagonal cycles associated with a trito-non-ordinary triple product of the form $f^B \times g^B \times \mathbf{h}^B$, where $f^B$ (resp. $g^B$) is a $p$-ordinary (resp. non-ordinary) eigenform on an indefinite quaternion algebra $B_{/\mathbb{Q}}$ of weight $2$, and $\mathbf{h}^B$ is a primitive Hida ($p$-ordinary) family. When $B=\mathrm{M}_2(\mathbb{Q})$ is split and $\mathbf{h}=\mathbf{h}^B$ has CM by an imaginary quadratic field, this allows us to develop the signed anticyclotomic Iwasawa theory for the base change $\mathrm{BC}_{K/\mathbb{Q}}(\pi_f)\times \mathrm{BC}_{K/\mathbb{Q}}(\pi_g)\times \psi$, where $\psi$ is a Hecke character of $K$. We formulate a signed Perrin-Riou-style Iwasawa main conjecture in this setting, and obtain a result on one inclusion in this conjecture. Our methods also allow us to extend Hsieh's construction of the balanced triple-product $p$-adic $L$-function to the trito-non-ordinary scenario, and to define its signed counterparts.