I discuss some algorithms for computing the zeta function of an algebraic variety over a finite field which are based upon rigid cohomology. Two distinct approaches are illustrated with a worked example.
Je présente quelques algorithmes pour calculer la fonction zêta d’une variété algébrique sur un corps fini qui sont basés sur la cohomologie rigide. Deux méthodes distinctes sont élaborées à l’aide d’un exemple.
@article{JTNB_2005__17_1_169_0, author = {Alan G.B. Lauder}, title = {Rigid cohomology and $p$-adic point counting}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {169--180}, publisher = {Universit\'e Bordeaux 1}, volume = {17}, number = {1}, year = {2005}, doi = {10.5802/jtnb.484}, mrnumber = {2152218}, zbl = {1087.14020}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.484/} }
TY - JOUR AU - Alan G.B. Lauder TI - Rigid cohomology and $p$-adic point counting JO - Journal de théorie des nombres de Bordeaux PY - 2005 SP - 169 EP - 180 VL - 17 IS - 1 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.484/ DO - 10.5802/jtnb.484 LA - en ID - JTNB_2005__17_1_169_0 ER -
%0 Journal Article %A Alan G.B. Lauder %T Rigid cohomology and $p$-adic point counting %J Journal de théorie des nombres de Bordeaux %D 2005 %P 169-180 %V 17 %N 1 %I Université Bordeaux 1 %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.484/ %R 10.5802/jtnb.484 %G en %F JTNB_2005__17_1_169_0
Alan G.B. Lauder. Rigid cohomology and $p$-adic point counting. Journal de théorie des nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 169-180. doi : 10.5802/jtnb.484. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.484/
[1] P. Bachmann, Zur Theory von Jacobi’s Kettenbruch-Algorithmen. J. Reine Angew. Math. 75 (1873), 25–34.
[2] E. Bombieri, On exponential sums in finite fields II. Invent. Math. 47 (1978), 29–39. | MR | Zbl
[3] J-P. Dedieu, Newton’s method and some complexity aspects of the zero-finding problem. In “Foundations of Computational Mathematics”, (R.A. DeVore, A. Iserles, E. Suli), LMS Lecture Note Series 284, Cambridge University Press, 2001, 45–67. | MR | Zbl
[4] B. Dwork, On the rationality of the zeta function of an algebraic variety. Amer. J. Math. 82 (1960), 631–648. | MR | Zbl
[5] B. Dwork, On the zeta function of a hypersurface II. Ann. Math. (2) 80 (1964), 227–299. | MR | Zbl
[6] N. Elkies, Elliptic and modular curves over finite fields and related computational issues. In “Computational perspectives in number theory: Proceedings of a conference in honour of A.O.L. Atkin” , (D.A. Buell and J.T. Teitelbaum), American Mathematical Society International Press 7 (1998), 21–76. | MR | Zbl
[7] M.D. Huang, D. Ierardi, Counting points on curves over finite fields. J. Symbolic Comput. 25 (1998), 1–21. | MR | Zbl
[8] K. Kedlaya, Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology. Journal of the Ramanujan Mathematical Society 16 (2001), 323–338. | MR | Zbl
[9] K. Kedlaya, Finiteness of rigid cohomology with coefficients, preprint 2002. | MR | Zbl
[10] A.G.B Lauder, Deformation theory and the computation of zeta functions, Proceedings of the London Mathematical Society 88 (3) (2004), 565-602. | MR | Zbl
[11] A.G.B. Lauder, Counting solutions to equations in many variables over finite fields, Foundations of Computational Mathematics 4 (3) (2004), 221-267. | MR | Zbl
[12] A.G.B. Lauder, D. Wan, Counting points on varieties over finite fields of small characteristic. To appear in Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography (Mathematical Sciences Research Institute Publications), J.P. Buhler and P. Stevenhagen (eds), Cambridge University Press. Available at: http://www.maths.ox.ac.uk/lauder/
[13] J. Pila, Frobenius maps of abelian varieties and finding roots of unity in finite fields. Math. Comp. 55 No. 192 (1990), 745–763. | MR | Zbl
[14] R. Schoof, Elliptic curves over finite fields and the computation of square roots mod . Math. Comp. 44 no. 170 (1985), 483–494. | MR | Zbl
[15] N. Tsuzuki, Bessel F-isocrystals and an algorithm for computing Kloosterman sums, preprint 2003.
[16] D. Wan, Algorithmic theory of zeta functions. To appear in Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography (Mathematical Sciences Research Institute Publications), J.P. Buhler and P. Stevenhagen (eds), Cambridge University Press. Available at: http://www.math.uci.edu/~dwan/preprint.html
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