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

Christine Noot-Huyghe, Nathalie Wach

Publication date: 7 June 2013

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.

Mathematics Subject Classification ID

Integral representations (11S23) de Rham cohomology and algebraic geometry (14F40) (p)-adic cohomology, crystalline cohomology (14F30)

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