p-adic q-expansion principles on unitary Shimura varieties

TL;DR

This paper establishes a new p-adic q-expansion principle for automorphic forms on unitary Shimura varieties by using Serre-Tate expansions, extending known results to arbitrary signatures.

Contribution

It introduces a p-adic q-expansion principle for unitary groups of arbitrary signature using Serre-Tate expansions, filling a gap in the literature.

Findings

Proves vanishing theorems for p-adic automorphic forms.

Establishes the equivalence between zero coefficients in Serre-Tate expansions and form vanishing.

Provides an expository overview complementing Hida's work.

Abstract

We formulate and prove certain vanishing theorems for p-adic automorphic forms on unitary groups of arbitrary signature. The p-adic q-expansion principle for p-adic modular forms on the Igusa tower says that if the coefficients of (sufficiently many of) the q-expansions of a p-adic modular form f are zero, then f vanishes everywhere on the Igusa tower. There is no p-adic q-expansion principle for unitary groups of arbitrary signature in the literature. By replacing q-expansions with Serre-Tate expansions (expansions in terms of Serre-Tate deformation coordinates) and replacing modular forms with automorphic forms on unitary groups of arbitrary signature, we prove an analogue of the p-adic q-expansion principle. More precisely, we show that if the coefficients of (sufficiently many of) the Serre-Tate expansions of a p-adic automorphic form f on the Igusa tower (over a unitary Shimura variety) are zero, then f vanishes identically on the Igusa tower. This paper also contains a substantial expository component. In particular, the expository component serves as a complement to Hida's extensive work on p-adic automorphic forms.