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

Abstract

We 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 $r\ge 2$, we show that the cohomology exhibits instability for large genus, whereas it remains stable for $r=0$ or $r=1$. As a corollary, we prove that the symplectic Prym representation associated with any finite abelian regular cover of a non-closed finite-type surface is infinitesimally rigid.