Counting the number of points of curves in a family

Abstract

The zeta function of an algebraic variety over a finite field k is an exponential sum involving the number of points on the variety over all finite extensions of k. One would like to compute zeta functions in an efficient way. This thesis discusses a method for computing zeta functions of smooth projective hypersurfaces called the deformation method. This method involves embedding the hypersurface in a family of hypersurfaces, containing a hypersurface for which we know the action of Frobenius on cohomology. The action of Frobenius satisfies a p-adic differential equation. Using this we can compute the zeta function of the hypersurface we started with. In this thesis, we formulate the deformation method for families in many variables. We investigate if we can use the deformation method for the universal curve of genus 3. There turns out to be an explicit condition for when the deformation method fails, and we try to understand this condition. The condition is very unnatural in the sense that it depends on the choice of coordinates.