Rigid cohomology and p-adic point counting
Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 169-180.

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.

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.

@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},
     zbl = {1087.14020},
     mrnumber = {2152218},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.484/}
}
Alan G.B. Lauder. Rigid cohomology and $p$-adic point counting. Journal de Théorie des Nombres de Bordeaux, Tome 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 506272 | Zbl 0396.14001

[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 1836614 | Zbl 0978.65048

[4] B. Dwork, On the rationality of the zeta function of an algebraic variety. Amer. J. Math. 82 (1960), 631–648. | MR 140494 | Zbl 0173.48501

[5] B. Dwork, On the zeta function of a hypersurface II. Ann. Math. (2) 80 (1964), 227–299. | MR 188215 | Zbl 0173.48601

[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 1486831 | Zbl 0915.11036

[7] M.D. Huang, D. Ierardi, Counting points on curves over finite fields. J. Symbolic Comput. 25 (1998), 1–21. | MR 1600606 | Zbl 0919.11046

[8] K. Kedlaya, Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology. Journal of the Ramanujan Mathematical Society 16 (2001), 323–338. | MR 1877805 | Zbl 1066.14024

[9] K. Kedlaya, Finiteness of rigid cohomology with coefficients, preprint 2002. | MR 2239343 | Zbl 05049027

[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 2044050 | Zbl 02094148

[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 2078663 | Zbl 1076.11040

[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: | Zbl pre05532114

[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 1035941 | Zbl 0724.11070

[14] R. Schoof, Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp. 44 no. 170 (1985), 483–494. | MR 777280 | Zbl 0579.14025

[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: