%0 Book Section
%T Point Counting on Non-Hyperelliptic Genus 3 Curves with Automorphism Group $\mathbb{Z} / 2 \mathbb{Z}$ using Monsky-Washnitzer Cohomology
%+ Institut de Mathématiques de Marseille (I2M)
%A Shieh, Yih-Dar
%Z 17 pages. Published in Algorithmic Arithmetic, Geometry, and Coding Theory, Contemporary Mathematics, vol. 637, Amer. Math. Soc., Providence, RI, 2015, pp. 173-189
%@ 978-1-4704-1461-0
%B Algorithmic Arithmetic, Geometry, and Coding Theory
%I Amer. Math. Soc., Providence, RI
%S Contemporary Mathematics
%V 637
%P 173--189
%8 2015
%D 2015
%Z 1603.00566
%R 10.1090/conm/637/12757
%Z 11G05; 11G40
%Z Mathematics [math]/Algebraic Geometry [math.AG]Book sections
%X 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 $C$.
%G English
%L hal-01321844
%U https://hal.science/hal-01321844
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-
%~ ANR