Higher congruence companion forms

TL;DR

This paper investigates the existence of companion forms for Hilbert modular newforms modulo prime powers, using deformation theory and modularity lifting under certain image size conditions.

Contribution

It establishes conditions under which splitting of Galois representations implies the existence of companion forms modulo p^n.

Findings

Splitting of Galois representations leads to companion forms.

Uses deformation theory and modularity lifting techniques.

Results apply to p-ordinary Hilbert modular newforms.

Abstract

For a rational prime $p \geq 3$ we consider $p$-ordinary, Hilbert modular newforms $f$ of weight $k\geq 2$ with associated $p$-adic Galois representations $\rho_f$ and $\mod{p^n}$ reductions $\rho_{f,n}$. Under suitable hypotheses on the size of the image, we use deformation theory and modularity lifting to show that if the restrictions of $\rho_{f,n} $ to decomposition groups above $p$ split then $f$ has a companion form $g$ modulo $p^n$ (in the sense that $\rho_{f,n}\sim \rho_{g,n}\otimes\chi^{k-1}$).