Asymptotic values of modular multiplicities for GL_2

TL;DR

This paper investigates the asymptotic behavior of modular multiplicities in algebraic representations of GL_2 over finite extensions of Q_p, revealing dependence on dimension and central character, and applies findings to the Breuil-Mézard conjecture.

Contribution

It provides the first asymptotic analysis of modular multiplicities for GL_2 representations, connecting multiplicity behavior to representation dimension and central character.

Findings

Multiplicities depend asymptotically only on dimension and central character.

Explicit asymptotic values are computed for multiplicities related to the Breuil-Mézard conjecture.

The results advance understanding of modular representation theory of p-adic groups.

Abstract

We study the irreducible constituents of the reduction modulo p of irreducible algebraic representations V of Res_{K/Q_p} GL_2 for K a finite extension of Q_p. We show that asymptotically, the multiplicity of each constituent depends only on the dimension of V and the central character of its reduction modulo p. As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-M\'ezard conjecture.