Differential operators and families of automorphic forms on unitary groups of arbitrary signature

TL;DR

This paper extends the theory of $p$-adic families of automorphic forms to unitary groups of arbitrary signature, removing previous restrictions and constructing explicit $p$-adic measures and Eisenstein series.

Contribution

It adapts Katz's differential operator approach to unitary groups without $q$-expansions, enabling the construction of $p$-adic families and measures for all signatures.

Findings

Explicit description of differential operators on Serre-Tate expansions.

Construction of a $p$-adic measure valued in automorphic forms.

Relation of the measure to $p$-adic Eisenstein series.

Abstract

In the 1970's, Serre exploited congruences between $q$-expansion coefficients of Eisenstein series to produce $p$-adic families of Eisenstein series and, in turn, $p$-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to $p$-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on $q$-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack $q$-expansions when the signature is of the form $(a, b)$, $a\neq b$. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a $p$-adic measure taking values in the space of $p$-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit $p$-adic family of Eisenstein series. One application of our results is to the recently completed construction of $p$-adic $L$-functions for unitary groups by the first named author, Harris, Li, and Skinner.