Wojciech J. Gajda

Abstract:

Let K be the fraction field of a dvr R. Given a symmetric bilinear space V over K, and a group G acting by isometries on V we give necessary and sufficient criteria for V to contain a unimodular lattice stabilized by G. We sketch two applications to zeta functions of varieties over finite fields. In one direction, the theorem gives restrictions on the possible characteristic polynomials of Frobenius on the middle cohomology of a smooth projective variety of even dimension over a finite field. This application generalizes (and gives a more conceptual proof of) a theorem of Elsenhans and Jahnel. In the other direction, the theorem plays a crucial role in establishing the existence of K3 surfaces over finite fields with given zeta-function.