A proof of the Breuil-Schneider conjecture in the indecomposable case

TL;DR

This paper proves the Breuil-Schneider conjecture for invariant norms on certain locally algebraic representations of reductive groups, extending previous results to a broader class using trace formulas and $p$-adic modular forms.

Contribution

It provides a proof of the conjecture in the indecomposable case for all connected reductive groups, generalizing prior work on $ ext{GL}(n)$.

Findings

Established the existence of invariant norms in the indecomposable case.

Extended the conjecture's proof from $ ext{GL}(n)$ to all connected reductive groups.

Connected the norm construction to classical $p$-adic modular forms.

Abstract

This paper contains a proof of a conjecture of Breuil and Schneider, on the existence of an invariant norm on any locally algebraic representation of $\GL(n)$, with integral central character, whose smooth part is given by a generalized Steinberg representation. In fact, we prove the analogue for any connected reductive group $G$. This is done by passing to a global setting, using the trace formula for an $\R$-anisotropic model of $G$. The ultimate norm comes from classical $p$-adic modular forms.