Explicit crystalline lattices in rigid cohomology

TL;DR

This paper provides an explicit description of integral lattices in rigid cohomology, aiding in p-adic point counting algorithms by reducing computational precision for zeta function calculations.

Contribution

It introduces a method to explicitly describe integral lattices in rigid cohomology using logarithmic de Rham complexes, applicable to certain hypersurfaces and their quotients.

Findings

Explicit lattices expressed via global sections of twisted logarithmic de Rham complexes

Main theorem proven for smooth proper hypersurfaces with smooth hyperplane sections

Results help reduce p-adic precision needed for zeta function computations

Abstract

Motivated by applications in point counting algorithms using p-adic cohomology, we give an explicit description of integral lattices in rigid cohomology spaces that p-adically approximate logarithmic crystalline cohomology modules. These lattices are expressed in terms of the global sections of the twisted logarithmic de Rham complex. We prove the main theorem for smooth proper hypersurfaces with a smooth hyperplane section, then deduce the result for the quotient of such a pair in weighted projective space. We show how these results may be used to reduce the necessary p-adic precision with which one must work to compute zeta functions of varieties over finite fields.