Relative Rigid Cohomology and Deformation of Hypersurfaces

The purpose of this article is to present a practical approach to computing zeta functions of smooth projective hypersurfaces over finite fields which is based on relative rigid cohomology. The method presented is a reformulation of a computational idea due to Lauder, which in turn was inspired by a beautiful paper by Dwork. The algorithm of Lauder, though a theoretical breakthrough, does not appear to be of immediate practical use; in particular, the precision-loss estimates appear too coarse to be useful. We derive a practical algorithm by recasting the computational idea in the setting of rigid cohomology and undertaking a delicate analysis of the precision loss, which is essential to improve the run-time behaviour. This also clarifies the relation to earlier point counting methods based on Monsky-Washnitzer cohomology, and shows how the methods can be gainfully combined. We have implemented our algorithm in the programing language Magma and present some examples which we have computed.