$(\varphi,\Gamma)$-modules associ\'es aux courbes hyperelliptiques lisses

TL;DR

This paper presents methods to compute the $(,B3)$-module associated with hyperelliptic curves over finite fields, using crystalline cohomology lattices and the Deligne-Illusie morphism, advancing computational techniques in arithmetic geometry.

Contribution

It introduces new approaches to compute $(,3B3)$-modules for hyperelliptic curves, leveraging crystalline cohomology lattices and the Deligne-Illusie morphism, building on Kedlaya's and Wach's methods.

Findings

Computed $(,3B3)$-modules using crystalline cohomology lattices.

Provided an alternative method via the Deligne-Illusie morphism.

Enhanced computational techniques for hyperelliptic curves.

Abstract

In 2003, Kedlaya gave an algorithm to compute the zeta function associated to a hyperelliptic curve over a finite field, by computing the rigid cohomology of the curve. Edixhoven remarked that it is actually possible to compute the crystalline cohomology of the curve, which is a lattice in the rigid cohomology. Following a method of Wach, we first explain how to use this lattice to compute the $(\varphi,\Gamma)$-module associated to an hyperelliptic curve. We also explain an alternative way to get the $(\varphi,\Gamma)$-module mod $p$ that relies on the Deligne-Illusie morphism.