Theta lifting and cohomology growth in p-adic towers

TL;DR

This paper investigates the growth of automorphic representations in p-adic towers using theta lifts, providing asymptotic results and verifying cases of a conjecture related to cohomology and automorphic forms.

Contribution

It introduces a method to analyze automorphic representation multiplicities in p-adic towers via theta lifts, extending previous surjectivity results and confirming parts of Sarnak and Xue's conjecture.

Findings

Sharp asymptotics for automorphic representations in p-adic towers

Verification of cases of Sarnak and Xue's conjecture

Application of theta lift surjectivity theorems

Abstract

We use the theta lift to study the multiplicity with which certain automorphic representations of cohomological type occur in a family of congruence covers of an arithmetic manifold. When the family of covers is a so-called `p-adic congruence tower' we obtain sharp asymptotics for the number of representations which occur as lifts. When combined with theorems on the surjectivity of the theta lift due to Howe and Li, and Bergeron, Millson and Moeglin, this allows us to verify certain cases of a conjecture of Sarnak and Xue.