Square root $p$-adic $L$-functions, I: Construction of a one-variable measure

TL;DR

This paper advances the construction of p-adic L-functions for automorphic forms on unitary groups, generalizing previous work to new settings involving Hida families and CM fields, with implications for special value formulas.

Contribution

It introduces a new measure construction for p-adic L-functions associated with automorphic representations on unitary groups, extending prior triple product methods to a broader context.

Findings

Constructed a one-variable p-adic measure for automorphic forms on unitary groups.

Extended previous triple product L-function techniques to new unitary group settings.

Utilized recent advances in p-adic Hodge theory and automorphic forms to support the construction.

Abstract

The Ichino-Ikeda conjecture, and its generalization to unitary groups by N. Harris, has given explicit formulas for central critical values of a large class of Rankin-Selberg tensor products. Although the conjecture is not proved in full generality, there has been considerable progress, especially for $L$-values of the form $L(1/2,BC(\pi) \times BC(\pi'))$, where $\pi$ and $\pi'$ are cohomological automorphic representations of unitary groups $U(V)$ and $U(V')$, respectively. Here $V$ and $V'$ are hermitian spaces over a CM field, $V$ of dimension $n$, $V'$ of codimension $1$ in $V$, and $BC$ denotes the twisted base change to $GL(n) \times GL(n-1)$. This paper contains the first steps toward generalizing the construction of my paper with Tilouine on triple product $L$-functions to this situation. We assume $\pi$ is a holomorphic representation and $\pi'$ varies in an ordinary Hida family (of antiholomorphic forms). The construction of the measure attached to $\pi$ uses recent work of Eischen, Fintzen, Mantovan, and Varma.