Point Counting on Non-Hyperelliptic Genus 3 Curves with Automorphism Group $\mathbb{Z} / 2 \mathbb{Z}$ using Monsky-Washnitzer Cohomology

TL;DR

This paper presents an efficient algorithm for computing the zeta function of non-hyperelliptic genus 3 curves with a specific automorphism group by leveraging Monsky-Washnitzer cohomology and quotient curve relations.

Contribution

It introduces a novel approach that splits the computation into parts, reducing complexity by exploiting the curve's automorphism group and quotient structure.

Findings

Faster computation of zeta functions for genus 3 curves.

Reduction of problem to point counting on an elliptic curve.

Improved efficiency over direct methods.

Abstract

We describe an algorithm to compute the zeta function of any non-hyperelliptic genus 3 plane curve $C$ over a finite field with automorphism group $G = \mathbb{Z} / 2 \mathbb{Z}$. This algorithm computes in the Monsky-Washnitzer cohomology of~the curve. Using the relation between the Monsky-Washnitzer cohomology of $C$ and its quotient $E := C/G$, the computation splits into 2 parts: one in a subspace of the Monsky-Washnitzer cohomology and a second which reduces to the point counting on an elliptic curve $E$. The former corresponds to the dimension $2$ abelian surface $\mathrm{ker}(\mathrm{Jac}(C) \rightarrow E)$, on which we can compute with lower precision and with matrices of smaller dimension. Hence we obtain a faster algorithm than working directly on the curve $C$.