Abstract
We present various improvements to the deformation method for computing the zeta function of smooth projective hypersurfaces over finite fields using \(p\)-adic cohomology. This includes new bounds for the \(p\)-adic and \(t\)-adic precisions required to obtain provably correct results and gains in the efficiency of the individual steps of the method. The algorithm that we thus obtain has lower time and space complexities than existing methods. Moreover, our implementation is more practical and can be applied more generally, which we illustrate with examples of generic quintic curves and quartic surfaces.
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Notes
This implementation is available at https://github.com/SPancratz/deformation.
References
T.G. Abbott, K.S. Kedlaya, and D. Roe. Bounding Picard numbers of surfaces using \(p\)-adic cohomology. In Arithmetic, Geometry, and Coding Theory (AGCT-10), 2006.
P. Berthelot. Géométrie rigide et cohomologie des variétés algébriques de caractéristique \(p\). Mém. Soc. Math. Fr. (N.S.), 23, 1986.
E. Bombieri. On exponential sums in finite fields. Invent. Math., 88: 71–105, 1966.
R.P. Brent. The complexity of multiple-precision arithmetic. In R.S. Anderssen and R.P. Brent, editors, The Complexity of Computational Problem Solving, pages 126–165. University of Queensland Press, 1976.
P. Deligne. La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math., 43, 1974.
P. Deligne and N.M. Katz. SGA 7 II, Groupes de Monodromie en Géométrie Algébrique. Springer Verlag, 1973.
B. Dwork. On the zeta function of a hypersurface I. Publication Mathématiques de l’IHÉS, 12: 5–68, 1962.
B. Dwork. On the zeta function of a hypersurface II. The Annals of Mathematics, Second Series, 80 (2): 227–299, 1964.
J.Y. Étesse and B. Le Stum. Fonctions L associées aux F-isocristaux surconvergents. I. interprétation cohomologique. Math. Ann., 296, 1993.
R. Gerkmann. Relative rigid cohomology and deformation of hypersurfaces. Int. Math. Res. Pap. IMRP, 1: 67, 2007. ISSN 1687–3017.
P.A. Griffiths. On the periods of certain rational integrals. I, (resp. II). Ann. of Math. (2), 90 (3): 460–495 (resp. 496–541), November 1969.
W. Hart, S. Pancratz, A. Novocin, F. Johansson, and D. Harvey. FLINT: Fast Library for Number Theory - Version 2.2, June 2011. http://www.libflint.org.
H. Hubrechts. Fast arithmetic in unramified \(p\)-adic fields. Finite Fields Appl., 16, 2010.
N.M. Katz. On the differential equations satisfied by period matrices. Inst. Hautes Études Sci. Publ. Math., 1968.
K.S. Kedlaya. Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology. Journal of the Ramanujan Mathematical Society, 16, 2001.
K.S. Kedlaya. Search techniques for root-unitary polynomials, volume 463 of Contemp. Math., pages 71–81. Amer. Math. Soc., Providence, RI, 2008.
K.S. Kedlaya. p-adic Differential Equations. Cambridge Univ. Press, 2010.
K.S. Kedlaya. Effective \(p\)-adic cohomology for cyclic cubic threefolds, Computational Algebraic and Analytic Geometry. Contemporary Mathematics 572, American Mathematical Society, 2013.
K.S. Kedlaya and J. Tuitman. Effective bounds for Frobenius structures. Rendiconti del Seminario Matematico della Universitá di Padova, page 9, 2012.
W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theoretical computer science, 36: 309–317, 1985.
A.G.B. Lauder. Counting solutions to equations in many variables over finite fields. Foundations of Computational Mathematics, 4 (3): 221–267, 2004a.
A.G.B. Lauder. Deformation theory and the computation of zeta functions. Proc. London Math. Soc, 3: 565–602, 2004b.
A.G.B. Lauder. A recursive method for computing zeta functions of varieties. LMS J. Comp. Math., 9: 222–269, 2006.
A.G.B. Lauder. Degenerations and limit Frobenius structures in rigid cohomology. LMS J. Comput. Math., 14: 1–33, 2011.
A.G.B. Lauder and D. Wan. Counting points on varieties over finite fields of small characteristic. In Algorithmic number theory: lattices, number fields, curves and cryptography, volume 44 of Math. Sci. Res. Inst. Publ., pages 579–612. Cambridge Univ. Press, Cambridge, 2008.
B. Mazur. Frobenius and the Hodge filtration. Bull. Amer. Math. Soc., 78, 1972.
R. Schoof. Counting points on elliptic curves over finite fields, 18ièmes Journées Arithmétiques, Bordeaux 1993. Journal de Théorie des Nombres de Bordeaux, 7: 219–254, 1995.
Virginia Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd [extended abstract]. In STOC’12–Proceedings of the 2012 ACM Symposium on Theory of Computing, pages 887–898. ACM, New York, 2012.
Acknowledgments
Both authors were supported by the European Research Council (Grant 204083) and additionally the second author was supported by FWO—Vlaanderen. We would like to thank Alan Lauder for all his help and in particular for his comments and suggestions on earlier versions of this paper. Finally, we thank the anonymous referees for their comments and suggestions.
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Felipe Cucker.
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Pancratz, S., Tuitman, J. Improvements to the Deformation Method for Counting Points on Smooth Projective Hypersurfaces. Found Comput Math 15, 1413–1464 (2015). https://doi.org/10.1007/s10208-014-9242-8
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DOI: https://doi.org/10.1007/s10208-014-9242-8