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

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