Trito-non-ordinary Iwasawa theory of diagonal cycles

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.

47 pages, comments are very welcome