Improvements to the deformation method for counting points on smooth projective hypersurfaces

TL;DR

This paper enhances the deformation method for calculating zeta functions of smooth projective hypersurfaces over finite fields, improving efficiency, precision bounds, and practical applicability with new algorithms and implementations.

Contribution

It introduces new bounds and optimizations for the deformation method, resulting in a more efficient and practical algorithm for computing zeta functions of hypersurfaces.

Findings

Lower time and space complexities compared to existing methods

More practical implementation applicable to various examples

Improved bounds for p-adic and t-adic precisions

Abstract

We present various improvements to the deformation method for computing the zeta function of smooth projective hypersurfaces over finite fields using $p$-adic cohomology. This includes new bounds for the $p$-adic and $t$-adic precisions required to obtain provably correct results and gains in the efficiency of the individual steps of the method. The algorithm that we thus obtain has lower time and space complexities than existing methods. Moreover, our implementation is more practical and can be applied more generally, which we illustrate with examples of quintic curves and quartic surfaces.