A $p$-adic approach to the existence of level-raising congruences

TL;DR

This paper develops a $p$-adic method to establish level-raising congruences among automorphic representations and applies it to prove symmetric power functoriality for Hilbert modular forms, confirming the existence of certain lifts.

Contribution

It introduces a novel $p$-adic approach to level-raising and demonstrates the existence of symmetric power lifts for Hilbert modular forms up to the 25th power.

Findings

Constructed level-raising congruences for $p$-ordinary automorphic representations

Proved the existence of the $n$th symmetric power lift for odd $n$ up to 25

Applied the method to symmetric power functoriality for Hilbert modular forms

Abstract

We construct level-raising congruences between $p$-ordinary automorphic representations, and apply this to the problem of symmetric power functoriality for Hilbert modular forms. In particular, we prove the existence of the $n^\text{th}$ symmetric power lift of a Hilbert modular eigenform of regular weight for each odd integer $n = 1, 3, \dots, 25$.