$(\phi ,\Gamma )$-modules associés aux courbes hyperelliptiques lisses

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 $(\phi ,\Gamma )$-module associated to an hyperelliptic curve. We also explain an alternative way to get the $(\phi ,\Gamma )$-module mod $p$ that relies on the Deligne-Illusie morphism.

28 pages