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

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1124
Classification: 14F20, 14Q05, 14Q20
Keywords: Algebraic geometry, Algorithm, Curves, Étale cohomology
Jinbi Jin 1

1 The Netherlands
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Computation of \'etale cohomology on curves in single exponential time},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Jinbi Jin. Computation of étale cohomology on curves in single exponential time. Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 2, pp. 311-354. doi : 10.5802/jtnb.1124. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1124/

[1] Stuart J. Berkowitz On computing the determinant in small parallel time using a small number of processors, Inf. Process. Lett., Volume 18 (1984), pp. 147-150 | DOI | MR | Zbl

[2] Alexander Chistov Algorithm of polynomial complexity for factoring polynomials and finding the components of varieties in subexponential time, J. Sov. Math., Volume 34 (1986) no. 4, pp. 1838-1882 (Translated from Zap. Nauchn. Sem. S.-Petersburg, 137:124–188, 1984) | DOI | Zbl

[3] Computational aspects of modular forms and Galois representations (Jean-Marc Couveignes; Sebastiaan J. Edixhoven, eds.), Annals of Mathematics Studies, 176, Princeton University Press, 2011 | MR | Zbl

[4] Richard Dedekind; Heinrich Weber Theorie der algebraischen Funktionen einer Veränderlichen, J. Reine Angew. Math. (1882)

[5] Pierre Deligne Cohomologie étale, Lecture Notes in Mathematics, 569, Springer, 1977 | Zbl

[6] Pierre Deligne Le déterminant de la cohomologie, Current trends in arithmetical algebraic geometry (Contemporary Mathematics), Volume 67, American Mathematical Society, 1987, pp. 93-177 | DOI | Zbl

[7] Alicia Dickenstein; Noaï Fitchas; Marc Giusti; Carmen Sessa The membership problem for unmixed polynomial ideals is solvable in single exponential time, Discrete Appl. Math., Volume 33 (1991), pp. 73-94 | DOI | MR | Zbl

[8] Claus Diem On arithmetic and the discrete logarithm problem in class groups of curves, 2008 (Habilitation thesis) | Zbl

[9] Daniel Ferrand Un foncteur norme, Bull. Soc. Math. Fr., Volume 126 (1998), pp. 1-49 | DOI | Numdam | MR | Zbl

[10] Lei Fu Étale cohomology theory, Nankai Tracts in Mathematics, 14, World Scientific, 2015 | Zbl

[11] Ulrich Görtz; Torsten Wedhorn Algebraic geometry I. Schemes. With examples and exercises, Advanced Lectures in Mathematics, Vieweg+Teubner, 2010 | DOI | Zbl

[12] Alexander Grothendieck Éléments de Géométrie Algébrique II. Etude globale élémentaire de quelques classes de morphismes, Publ. Math., Inst. Hautes Étud. Sci., Volume 8 (1961), pp. 5-222 | Numdam

[13] Alexander Grothendieck Revêtements étales et groupe fondamental (SGA1), Lecture Notes in Mathematics, 224, Springer, 1971

[14] David Harvey Counting points on hyperelliptic curves in average polynomial time, Ann. Math., Volume 179 (2014) no. 2, pp. 783-803 | DOI | MR | Zbl

[15] Florian Hess Computing Riemann–Roch spaces in algebraic function fields and related topics, J. Symb. Comput., Volume 33 (2002) no. 4, pp. 425-445 | DOI | MR | Zbl

[16] Erich Kaltofen Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization, SIAM J. Comput., Volume 14 (1985) no. 2, pp. 469-489 | DOI | MR | Zbl

[17] Kiran S. Kedlaya Counting points on hyperelliptic curves using Monsky–Washnitzer cohomology, J. Ramanujan Math. Soc., Volume 16 (2001) no. 4, pp. 323-338 | MR | Zbl

[18] Kamal Khuri-Makdisi Asymptotically fast group operations on Jacobians of general curves (2004) (https://arxiv.org/abs/math/0409209v2) | Zbl

[19] Alan Lauder; Daqing Wan Counting points on varieties over finite fields of small characteristic, Algorithmic Number Theory (Mathematical Sciences Research Institute Publications), Volume 44, Cambridge University Press, 2008 | MR

[20] Max Lieblich Galois representations arising from p-divisible groups, Ph. D. Thesis, Harvard University (2000)

[21] David A. Madore; Fabrice Orgogozo Calculabilité de la cohomologie étale modulo , Algebra Number Theory, Volume 9 (2015) no. 7, pp. 1647-1739 | DOI | Zbl

[22] Bjorn Poonen; Damiano Testa; Ronald van Luijk Computing Néron–Severi groups and cycle class groups, Compos. Math., Volume 151 (2015) no. 4, pp. 713-734 | DOI | Zbl

[23] Stacks Project Authors Stacks Project, 2014 (http://stacks.math.columbia.edu)

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