On middle cohomology of special Artin–Schreier varieties and finite Heisenberg groups

Abstract

Abstract We introduce a certain Artin–Schreier scheme over a finite field associated to a pair of coprime integers ( m , n ) {(m,n)} with n ≥ 3 {n\geq 3} divisible by the characteristic of the base field, and study the middle étale cohomology group of it. If m is even, the variety admits actions of some finite Heisenberg groups. We study the middle cohomology as representations of the Heisenberg groups. If m is odd, we compute the Frobenius eigenvalues of it concretely. This affine scheme comes from the reduction of a certain affinoid in a Lubin–Tate space.