Prym Representations and Twisted Cohomology of the Mapping Class Group with Level Structures

In my thesis, I compute the twisted cohomology of the mapping class group with level structures with coefficients in the r-tensor powers of the Prym representations for any positive integer r. When r = 2, the cohomology turns out to be not stable when the genus is large, but it is stable when r is 0 or 1. As a corollary, we prove that the symplectic Prym representation of any finite abelian regular cover of a non-closed finite-type surface is infinitesimally rigid.