$p$-adic $L$-functions for $\mathrm U(2,1)\times\mathrm U(1,1)$

TL;DR

The paper develops a five-variable p-adic L-function attached to Hida families on the quasi-split unitary groups ${\mathrm U}(2,1)\times{\mathrm U}(1,1)$, interpolating a square root of central L-values in the shifted piano range and aligning with Coates–Perrin-Riou predictions. It introduces a novel p-adic theta operator $\boldsymbol{\Theta}_{\mathfrak p}^{\phi}$ acting on Fourier–Jacobi coefficients of Picard modular forms, and connects this operator to Shimura’s differential operators and Bannai–Kobayashi theory to obtain p-adic interpolation of L-values via a tight web of Fourier–Jacobi data, Ichino–Ikeda type period formulas, and Panchishkin filtrations. The construction relies on a refined geometric framework of Shimura varieties for unitary groups, Igusa towers, and Hida theory for ${\mathrm{GU}}(r,1)$, together with analytic inputs from the restriction of Picard modular forms and Rankin–Selberg/L-series comparisons. An explicit interpolation formula, incorporating modified Euler factors at $p$ and at the archimedean place, demonstrates compatibility with the conjectural shapes predicted by Coates–Perrin-Riou and related works, and the work yields a pathway to derived one-variable p-adic L-functions via specialization and the theta-operator framework. The results illuminate a robust arithmetic–analytic bridge for unitary groups and offer tools for accessing $p$-adic periods and central L-values in the diagonal unitary setting, with potential applications to $p$-adic IINH formulas and diagonal cycle conjectures.

Abstract

We construct the five-variable $p$-adic $L$-function attached to Hida families on $\mathrm U(2,1)\times\mathrm U(1,1)$, interpolating the square-root of Rankin-Selberg $L$-values in the \emph{shifted piano} range. Our construction relies on a new theta operator and its $p$-adic variation which plays a role analogous to the classical Ramanujan-Serre theta operator in Hida's $p$-adic Rankin-Selberg method. The interpolation formula, including the modified Euler factors at $p$ and at the real place, is consistent with the conjectural shape of $p$-adic $L$-functions predicted by Coates and Perrin-Riou.

Theorems & Definitions (184)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 2.1
• Definition 2.2
• Remark 2.3
• Definition 3.1
• Definition 3.2