Frobenius lifts and point counting for smooth curves

TL;DR

This paper presents an algorithm for computing the zeta-function of smooth curves over finite fields using Frobenius lifts and cohomology, extending existing methods to more general cases.

Contribution

It introduces a new algorithm leveraging Frobenius lifts and Serre duality for point counting on smooth curves, generalizing previous hyperelliptic-specific approaches.

Findings

Algorithm has softly cubic complexity in the field degree

Method extends Kedlaya's algorithm beyond hyperelliptic curves

Efficient local computation of cup products and Frobenius action

Abstract

We describe an algorithm to compute the zeta-function of a proper, smooth curve over a finite field, when the curve is given together with some auxiliary data. Our method is based on computing the matrix of the action of a semi-linear Frobenius on the first cohomology group of the curve by means of Serre duality. The cup product involved can be computed locally, after first computing local expansions of a globally defined lift of Frobenius. The resulting algorithm's complexity is softly cubic in the field degree, which is also the case with Kedlaya's algorithm in the hyperelliptic case.