Artin L-functions of diagonal hypersurfaces and generalized hypergeometric functions over finite fields

Abstract

We compute the Artin $L$-function of a diagonal hypersurface $D_{\lambda }$ over a finite field associated to a character of a finite group acting on $D_{\lambda }$, and under some condition, express it in terms of hypergeometric functions and Jacobi sums over the finite field. As an application, we derive certain relations among hypergeometric functions over different finite fields from the Grothendieck-Lefschetz trace formula.