Relative Rigid Cohomology and Point Counting on Families of Elliptic Curves

This article describes a method for the computation of the zeta functions of elliptic curves contained in algebraic families by evaluating p-adic meromorphic functions. These functions occur as solutions of a p-adic differential equation, which is derived from the Gauss–Manin connection on the relative de Rham cohomology of a p-adic lift. Stated in terms of rigid cohomology, we work with an F-isocrystal on the base scheme P¹\{0, 1,∞} of the family. This work combines the idea of Lauder’s deformation theoretic approach based on Dwork homology and Kedlaya’s algorithm for counting points on a particular curve which uses Monsky–Washnitzer cohomology.