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

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 $\ker (\mathrm{Jac}(C)\to E)$, 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 $C$.

17 pages. Published in Algorithmic Arithmetic, Geometry, and Coding Theory, Contemporary Mathematics, vol. 637, Amer. Math. Soc., Providence, RI, 2015, pp. 173-189

References in corpus (1)

• Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology