Computation of étale cohomology on curves in single exponential time
Jinbi Jin1
1The Netherlands
Journal de théorie des nombres de Bordeaux, Tome 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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1124
Classification : 14F20, 14Q05, 14Q20
Keywords: Algebraic geometry, Algorithm, Curves, Étale cohomology

Jinbi Jin  1

1 The Netherlands
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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
@article{JTNB_2020__32_2_311_0,
     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},
     year = {2020},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {2},
     doi = {10.5802/jtnb.1124},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1124/}
}
TY  - JOUR
AU  - Jinbi Jin
TI  - Computation of étale cohomology on curves in single exponential time
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2020
SP  - 311
EP  - 354
VL  - 32
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1124/
DO  - 10.5802/jtnb.1124
LA  - en
ID  - JTNB_2020__32_2_311_0
ER  - 
%0 Journal Article
%A Jinbi Jin
%T Computation of étale cohomology on curves in single exponential time
%J Journal de théorie des nombres de Bordeaux
%D 2020
%P 311-354
%V 32
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1124/
%R 10.5802/jtnb.1124
%G en
%F JTNB_2020__32_2_311_0

[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 | Numdam | DOI | 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 ℓ, 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)

Cité par Sources :