## Number Theory Seminar

I will begin this talk by reviewing the Eichler-Shimura isomorphism between the space of cusp forms of weight k and level N, and a certain cohomology group. Shimura called this cohomology group as parabolic cohomology. In the context of automorphic forms on higher groups, such a cohomology group takes the form of cuspidal cohomology. The existence of cusp forms of prescribed weight then may be reformulated into the nonvanishing of cuspidal cohomology with prescribed coefficients. I will review the relevant definitions to state the nonvanishing problem for cuspidal cohomology for GL(n) over a number field F. After surveying the state of the art, I will discuss some recent results, obtained in joint work with my student Darshan Nasit, settling this problem when F is Galois over its maximal totally real subfield.