Trito-non-ordinary Iwasawa theory of diagonal cycles
Raúl Alonso, Kâzım Büyükboduk, Antonio Cauchi, Antonio Lei
TL;DR
This work constructs and analyzes signed diagonal cycles for the trito-non-ordinary triple product $f^B \times g^B \times \mathbf{h}^B$, enabling unbounded and signed balanced $p$-adic $L$-functions in the anticyclotomic setting. By combining Loeffler–Spherical diagonal cycles, CM Hida theory, and patching, the authors build Euler systems compatible with anticyclotomic twists over an imaginary quadratic field $K$ and formulate a signed Perrin–Riou–style main conjecture, proving one containment. They extend Hsieh’s balanced triple-product $p$-adic $L$-function to the trito-non-ordinary case and define its signed variants, connecting diagonal cycles with $p$-adic $L$-interpolation. The results yield a concrete pathway to relating Euler systems to Selmer groups and to deriving rank-one Iwasawa-theoretic information for the base-changed Rankin–Selberg product, with explicit norm relations and local-condition verifications. Overall, the paper advances signed Iwasawa theory for non-ordinary and ordinary factors in a unified triple-product framework and highlights potential bipartite Euler-system structures.
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 $\mathrm{BC}_{K/\mathbb{Q}}(π_f)\times \mathrm{BC}_{K/\mathbb{Q}}(π_g)\times ψ$, where $ψ$ 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.