Artin–Schreier varieties with actions of finite orthogonal groups and Weil representations

Abstract

We study Artin–Schreier varieties with natural actions of finite orthogonal groups. We compute the characters of the cohomology groups of these varieties as representations of the groups. Our main ingredient is Shinoda’s character formulae on Weil representations for symplectic groups. We give trace formulae for projective hypersurfaces with actions of finite orthogonal groups generalizing Fermat hypersurfaces.