$p$-adic families of automorphic forms in the $\mu$-ordinary setting

TL;DR

This paper develops a theory of $p$-adic automorphic forms on unitary groups that interpolates in families over the $$-ordinary locus, extending Hida's theory to more primes and constructing $p$-adic families using differential operators.

Contribution

It generalizes Hida's $p$-adic automorphic forms theory to the $$-ordinary setting, removing the splitting condition on $p$, and introduces new differential operators for constructing $p$-adic families.

Findings

Framework applies to all primes not ramifying in the reflex field

Construction of $p$-adic families using new differential operators

Extension of Hida's theory beyond the ordinary locus

Abstract

We develop a theory of $p$-adic automorphic forms on unitary groups that allows $p$-adic interpolation in families and holds for all primes $p$ that do not ramify in the reflex field $E$ of the associated unitary Shimura variety. If the ordinary locus is nonempty (a condition only met if $p$ splits completely in $E$), we recover Hida's theory of $p$-adic automorphic forms, which is defined over the ordinary locus. More generally, we work over the $\mu$-ordinary locus, which is open and dense. By eliminating the splitting condition on $p$, our framework should allow many results employing Hida's theory to extend to infinitely many more primes. We also provide a construction of $p$-adic families of automorphic forms that uses differential operators constructed in the paper. Our approach is to adapt the methods of Hida and Katz to the more general $\mu$-ordinary setting, while also building on papers of each author. Along the way, we encounter some unexpected challenges and subtleties that do not arise in the ordinary setting.