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

Yih-Dar Shieh (1)

1 I2M - Institut de Mathématiques de Marseille

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 $C$.

Domaines: Géométrie algébrique [math.AG]

Yih-Dar Shieh. Point Counting on Non-Hyperelliptic Genus 3 Curves with Automorphism Group $\mathbb{Z} / 2 \mathbb{Z}$ using Monsky-Washnitzer Cohomology. Algorithmic Arithmetic, Geometry, and Coding Theory, 637, Amer. Math. Soc., Providence, RI, pp.173--189, 2015, Contemporary Mathematics, 978-1-4704-1461-0. ⟨10.1090/conm/637/12757⟩. ⟨hal-01321844⟩