Congruences of algebraic automorphic forms and supercuspidal representations

TL;DR

This paper establishes congruences between automorphic forms with arbitrary and supercuspidal components for certain reductive groups over totally real fields, enabling new applications in Galois representations and extending known density results.

Contribution

It introduces a method to find congruences modulo p^n between automorphic forms with different local components for general reductive groups, generalizing previous results for GL(2).

Findings

Constructs new supercuspidal types with small compact open subgroups.

Proves congruences between automorphic forms with different local components.

Extends density of supercuspidal points to general reductive groups.

Abstract

Let $G$ be a connected reductive group over a totally real field $F$ which is compact modulo center at archimedean places. We find congruences modulo an arbitrary power of p between the space of arbitrary automorphic forms on $G(\mathbb A_F)$ and that of automorphic forms with supercuspidal components at p, provided that p is larger than the Coxeter number of the absolute Weyl group of $G$. We illustrate how such congruences can be applied in the construction of Galois representations. Our proof is based on type theory for representations of p-adic groups, generalizing the prototypical case of GL(2) in [arXiv:1506.04022, Section 7] to general reductive groups. We exhibit a plethora of new supercuspidal types consisting of arbitrarily small compact open subgroups and characters thereof. We expect these results of independent interest to have further applications. For example, we extend the result by Emerton--Pa\v{s}k\=unas on density of supercuspidal points from definite unitary groups to general $G$ as above.