Level raising for p-adic Hilbert modular forms

TL;DR

This paper extends level raising results to overconvergent p-adic automorphic forms over totally real fields, showing how local Galois representations can be deformed within families of eigenforms.

Contribution

It generalizes classical level raising theorems to the setting of overconvergent p-adic automorphic forms for quaternion algebras over totally real fields, with explicit examples and implications for Leopoldt's conjecture.

Findings

Overconvergent eigenforms with specific local Galois representations can be deformed into families with different local properties.

Explicit examples of p-adic automorphic forms satisfying the new level raising conditions.

Connections to Leopoldt's conjecture and local-global compatibility in Hilbert modular forms.

Abstract

This paper generalises previous work of the author to the setting of overconvergent $p$-adic automorphic forms for a definite quaternion algebra over a totally real field. We prove results which are analogues of classical `level raising' results in the theory of mod $p$ modular forms. Roughly speaking, we show that an overconvergent eigenform whose associated local Galois representation at some auxiliary prime $\l$ is (a twist of) a direct sum of trivial and cyclotomic characters lies in a family of eigenforms whose local Galois representation at $\l$ is generically (a twist of) a ramified extension of trivial by cyclotomic. We give some explicit examples of $p$-adic automorphic forms to which our results apply, and give a general family of examples whose existence would follow from counterexamples to the Leopoldt conjecture for totally real fields. These results also play a technical role in other work of the author on the problem of local--global compatibility at Steinberg places for Hilbert modular forms of partial weight one.