Reduction mod p of Cuspidal Representations of GL(2,q) and Symmetric Powers

TL;DR

This paper constructs integral models for cuspidal representations of GL(2,q) and explores their reduction mod p, linking representation theory with crystalline cohomology and modular forms.

Contribution

It introduces new integral models for cuspidal representations of GL(2,q) derived from crystalline cohomology, extending previous work on mod p modular forms.

Findings

Integral models correspond to cokernels of Serre's differential operator.

Reduction mod p relates to crystalline cohomology of a specific algebraic curve.

Extension of cohomological Hasse invariant operator to mod p modular forms.

Abstract

We show the existence of integral models for cuspidal representations of GL(2,q), whose reduction modulo p can be identified with the cokernel of a differential operator on F_{q}[X,Y] defined by J-P. Serre. These integral models come from the crystalline cohomology of the projective curve XY^{q}-X^{q}Y-Z^{q+1}=0. As an application, we can extend a construction of C. Khare and B. Edixhoven (2003) giving a cohomological analogue of the Hasse invariant operator acting on spaces of modp modular forms for GL(2).