$p$-adic $L$-functions for $P$-ordinary Hida families on unitary groups

TL;DR

This paper constructs a $p$-adic $L$-function for $P$-ordinary Hida families on unitary groups, incorporating Schneider-Zink types to handle higher ramification and enabling interpolation of Fourier coefficients and special $L$-values.

Contribution

It introduces a novel approach using Schneider-Zink types for the Levi quotient to study $p$-adic automorphic forms with higher ramification, advancing the construction of $p$-adic $L$-functions.

Findings

Construction of a $p$-adic $L$-function for $P$-ordinary Hida families.

Interpolation of Fourier coefficients into a $p$-adic Eisenstein measure.

Reinterpretation of the doubling method as a pairing for evaluating $p$-adic $L$-functions.

Abstract

We construct a $p$-adic $L$-function for $P$-ordinary Hida families of cuspidal automorphic representations on a unitary group $G$. The main new idea of our work is to incorporate the theory of Schneider-Zink types for the Levi quotient of $P$, to allow for the possibility of higher ramification at primes dividing $p$, into the study of ($p$-adic) modular forms and automorphic representations on $G$. For instance, we describe the local structure of such a $P$-ordinary automorphic representation $\pi$ at $p$ using these types, allowing us to analyze the geometry of $P$-ordinary Hida families. Furthermore, these types play a crucial role in the construction of certain Siegel Eisenstein series designed to be compatible with such Hida families in two specific ways : Their Fourier coefficients can be $p$-adically interpolated into a $p$-adic Eisenstein measure on $d+1$ variables and, via the doubling method of Garrett and Piatetski--Shapiro-Rallis, the corresponding zeta integrals yield special values of standard $L$-functions. Here, $d$ is the rank of the Levi quotient of $P$. Lastly, the doubling method is reinterpreted algebraically as a pairing between modular forms on $G$, whose nebentype are types, and viewed as the evaluation of our $p$-adic $L$-function at classical points of a $P$-ordinary Hida family.