Level lowering for GU(1,2)

TL;DR

This paper extends Mazur's level lowering principle to automorphic representations of the group GU(1,2), demonstrating that certain Galois representations with Steinberg ramification can also be realized unramified at the same prime.

Contribution

It proves a new level lowering result for Galois representations associated with automorphic forms on GU(1,2), analogous to Mazur's principle for modular forms.

Findings

Galois representations with Steinberg ramification can be realized unramified at the same prime.

The result applies to automorphic representations of the unitary similitude group GU(1,2).

Provides a new tool for understanding the ramification behavior of automorphic Galois representations.

Abstract

Mazur's principle gives a criterion under which an irreducible mod $\ell$ Galois representation arising from a modular form of level $Np$ (with $p$ prime to $N$) can also arise from a modular form of level $N.$ We prove an analogous result showing that a mod $\ell$ Galois representation arising from a stable cuspidal automorphic representation of the unitary similitude group $G=\mathrm{GU}(1,2)$ which is Steinberg at an inert prime $p$ can also arise from an automorphic representation of $G$ that is unramified at $p$.