Computation of étale cohomology on curves in single exponential time
Journal de Théorie des Nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 311-354.

Dans ce texte, on décrit un algorithme calculant, pour une courbe lisse et connexe X sur un corps k et un faisceau localement constant de groupes abéliens de torsion inversible dans k, le premièr groupe de cohomologie étale H 1 (X k sep ,ét ,𝒜) et le premièr groupe de cohomologie étale à support propre H c 1 (X k sep ,ét ,𝒜) comme ensembles de torseurs.

La complexité arithmétique de cet algorithme est exponentielle en n logn , p a (X), et p a (𝒜), où p a (X) est le genre arithmétique de la complétion normale de X sur k, p a (𝒜) est le genre arithmétique de la complétion normale de la courbe Y répresentant le faisceau 𝒜, et n est le degré de Y sur X.

L’algorithme passe par le calcul d’un schéma en groupoïdes classifiant les 𝒜-torseurs étales avec quelques structures additionnelles rigidifiantes.

In this paper, we describe an algorithm that, for a smooth connected curve X over a field k, a finite locally constant sheaf 𝒜 on X ét of abelian groups of torsion invertible in k, computes the first étale cohomology H 1 (X k sep ,ét ,𝒜) and the first étale cohomology with proper support H c 1 (X k sep ,ét ,𝒜) as sets of torsors.

The complexity of this algorithm is exponential in n logn , p a (X), and p a (𝒜), where p a (X) is the arithmetic genus of the normal completion of X, p a (𝒜) is the arithmetic genus of the normal completion Y of the smooth curve representing 𝒜, and n is the degree of Y over X.

The computation in this algorithm is done via the computation of a groupoid scheme classifying the 𝒜-torsors with some extra rigidifying data.

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DOI : https://doi.org/10.5802/jtnb.1124
Classification : 14F20,  14Q05,  14Q20
Mots clés : Algebraic geometry, Algorithm, Curves, Étale cohomology
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     author = {Jinbi Jin},
     title = {Computation of \'etale cohomology on curves in single exponential time},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {311--354},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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     doi = {10.5802/jtnb.1124},
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     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1124/}
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Jinbi Jin. Computation of étale cohomology on curves in single exponential time. Journal de Théorie des Nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 311-354. doi : 10.5802/jtnb.1124. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1124/

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