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).