Reduction mod p of cuspidal representations of GL2(Fpn) and symmetric powers
We show the existence of integral models for cuspidal representations of GL2(Fq), whose reduction modulo p can be identified with the cokernel of a differential operator on Fq[X,Y] defined by J.-P. Serre. These integral models come from the crystalline cohomology of the projective curve XYq−XqY−Zq+1=0. 
 
As an application, we can extend a construction of C. Khare and B. Edixhoven (2003) [5] giving a cohomological analogue of the Hasse invariant operator acting on spaces of mod p modular forms for GL2.