Computing periods of hypersurfaces
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Abstract:
We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of period integrals to an initial value problem for ordinary differential equations of Picard–Fuchs type. In this way, the periods can be computed to extreme precision in order to study their arithmetic properties. The initial conditions are obtained by an exact determination of the cohomology pairing on Fermat hypersurfaces with respect to a natural basis.References
- V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from the Russian by Hugh Porteous; Translation revised by the authors and James Montaldi. MR 966191, DOI 10.1007/978-1-4612-3940-6
- Emil Artin, The gamma function, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York-Toronto-London, 1964. Translated by Michael Butler. MR 0165148
- D. H. Bailey and J. M. Borwein, High-precision numerical integration: progress and challenges, J. Symbolic Comput. 46 (2011), no. 7, 741–754. MR 2795208, DOI 10.1016/j.jsc.2010.08.010
- Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler, Numerically solving polynomial systems with Bertini, Software, Environments, and Tools, vol. 25, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. MR 3155500
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- Paul Breiding and Sascha Timme, Homotopycontinuation.jl - a package for solving systems of polynomial equations in julia (2018), available at arXiv:1711.10911.
- Nils Bruin, Jeroen Sijsling, and Alexandre Zotine, Numerical computation of endomorphism rings of Jacobians, ANTS 2018 (2018).
- James A. Carlson and Phillip A. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli problem, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 51–76. MR 605336
- François Charles, On the Picard number of K3 surfaces over number fields, Algebra Number Theory 8 (2014), no. 1, 1–17. MR 3207577, DOI 10.2140/ant.2014.8.1
- L. Chua, M. Kummer, and B. Sturmfels, Schottky algorithms: Classical meets tropical, Math. Comp. Posted on December 27, 2018, DOI 10.1090/mcom/3406 (to appear in print).
- Frédéric Chyzak, An extension of Zeilberger’s fast algorithm to general holonomic functions, Discrete Math. 217 (2000), no. 1-3, 115–134 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR 1766263, DOI 10.1016/S0012-365X(99)00259-9
- C. Herbert Clemens and Phillip A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. MR 302652, DOI 10.2307/1970801
- David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. MR 1677117, DOI 10.1090/surv/068
- Bernard Deconinck and Mark van Hoeij, Computing Riemann matrices of algebraic curves, Phys. D 152/153 (2001), 28–46. Advances in nonlinear mathematics and science. MR 1837895, DOI 10.1016/S0167-2789(01)00156-7
- Bernard Deconinck and Matthew S. Patterson, Computing with plane algebraic curves and Riemann surfaces: the algorithms of the Maple package “algcurves”, Computational approach to Riemann surfaces, Lecture Notes in Math., vol. 2013, Springer, Heidelberg, 2011, pp. 67–123. MR 2905611, DOI 10.1007/978-3-642-17413-1_{2}
- Alex Degtyarev and Ichiro Shimada, On the topology of projective subspaces in complex Fermat varieties, J. Math. Soc. Japan 68 (2016), no. 3, 975–996. MR 3523534, DOI 10.2969/jmsj/06830975
- Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982. MR 654325
- The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 8.1), 2017. http://www.sagemath.org.
- Bernard Dwork, On the zeta function of a hypersurface, Inst. Hautes Études Sci. Publ. Math. 12 (1962), 5–68. MR 159823
- Andreas-Stephan Elsenhans and Jörg Jahnel, Real and complex multiplication on K3 surfaces via period integration (2018), available at arXiv:1802.10210v1.
- Jörg Frauendiener and Christian Klein, Computational approach to compact Riemann surfaces, Nonlinearity 30 (2017), no. 1, 138–172.
- Fritz Gesztesy and Helge Holden, Soliton equations and their algebro-geometric solutions. Vol. I, Cambridge Studies in Advanced Mathematics, vol. 79, Cambridge University Press, Cambridge, 2003. $(1+1)$-dimensional continuous models. MR 1992536, DOI 10.1017/CBO9780511546723
- Patrizia Gianni, Mika Seppälä, Robert Silhol, and Barry Trager, Riemann surfaces, plane algebraic curves and their period matrices, J. Symbolic Comput. 26 (1998), no. 6, 789–803. Symbolic numeric algebra for polynomials. MR 1662036, DOI 10.1006/jsco.1998.0240
- Phillip A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460-495; ibid. (2) 90 (1969), 496–541. MR 0260733, DOI 10.2307/1970746
- Benedict H. Gross, On the periods of abelian integrals and a formula of Chowla and Selberg, Invent. Math. 45 (1978), no. 2, 193–211. With an appendix by David E. Rohrlich. MR 480542, DOI 10.1007/BF01390273
- Samuel Grushevsky, The Schottky problem, Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press, Cambridge, 2012, pp. 129–164. MR 2931868
- Samuel Grushevsky and Martin Möller, Explicit formulas for infinitely many Shimura curves in genus 4, Asian J. Math. 22 (2018), no. 2, 381–390. MR 3824574, DOI 10.4310/ajm.2018.v22.n2.a12
- J. Guàrdia, Explicit geometry on a family of curves of genus 3, J. London Math. Soc. (2) 64 (2001), no. 2, 299–310. MR 1853452, DOI 10.1112/S0024610701002538
- Jordi Guàrdia, On the Torelli problem and Jacobian Nullwerte in genus three, Michigan Math. J. 60 (2011), no. 1, 51–65. MR 2785863, DOI 10.1307/mmj/1301586303
- Jonathan D. Hauenstein and Andrew J. Sommese, What is numerical algebraic geometry? [Foreword]. part 3, J. Symbolic Comput. 79 (2017), no. part 3, 499–507. MR 3563094, DOI 10.1016/j.jsc.2016.07.015
- Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175–232. MR 291177
- George R. Kempf, The equations defining a curve of genus $4$, Proc. Amer. Math. Soc. 97 (1986), no. 2, 219–225. MR 835869, DOI 10.1090/S0002-9939-1986-0835869-2
- Christoph Koutschan, A fast approach to creative telescoping, Math. Comput. Sci. 4 (2010), no. 2-3, 259–266. MR 2775992, DOI 10.1007/s11786-010-0055-0
- Pierre Lairez, Computing periods of rational integrals, Math. Comp. 85 (2016), no. 300, 1719–1752. MR 3471105, DOI 10.1090/mcom/3054
- Pierre Lairez and Emre Can Sertöz, A numerical transcendental method in algebraic geometry (2018), available at arXiv:1811.10634.
- Eduard Looijenga, Fermat varieties and the periods of some hypersurfaces, Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), Adv. Stud. Pure Math., vol. 58, Math. Soc. Japan, Tokyo, 2010, pp. 47–67. MR 2676157, DOI 10.2969/aspm/05810047
- Ronald van Luijk, K3 surfaces with Picard number one and infinitely many rational points, Algebra Number Theory 1 (2007), no. 1, 1–15. MR 2322921, DOI 10.2140/ant.2007.1.1
- Marc Mezzarobba, Rigorous multiple-precision evaluation of d-finite functions in sagemath (2016), available at arXiv:1607.01967v1.
- Michael B. Monagan, Keith O. Geddes, K. Michael Heal, George Labahn, Stefan M. Vorkoetter, James McCarron, and Paul DeMarco, Maple 10 Programming Guide, Maplesoft, Waterloo ON, Canada, 2005.
- H. Movasati, A course in Hodge theory, with emphasis in multiple integrals, to appear. [Online]. Available: w3.impa.br/~hossein/myarticles/hodgetheory.pdf.
- Sebastian Pancratz and Jan Tuitman, Improvements to the deformation method for counting points on smooth projective hypersurfaces, Found. Comput. Math. 15 (2015), no. 6, 1413–1464. MR 3413626, DOI 10.1007/s10208-014-9242-8
- Frédéric Pham, Formules de Picard-Lefschetz généralisées et ramification des intégrales, Bull. Soc. Math. France 93 (1965), 333–367 (French). MR 195868
- Bernhard Riemann, Zur theorie der abelschen funktionen für den fall p = 3, Gesam- melte mathematishce werke und wissenschaftlicher nachlass, 1876.
- Matthias Schütt, Picard numbers of quintic surfaces, Proc. Lond. Math. Soc. (3) 110 (2015), no. 2, 428–476. MR 3335284, DOI 10.1112/plms/pdu056
- Mika Seppälä, Computation of period matrices of real algebraic curves, Discrete Comput. Geom. 11 (1994), no. 1, 65–81. MR 1244890, DOI 10.1007/BF02573995
- Tetsuji Shioda, On the Picard number of a complex projective variety, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 303–321. MR 644520
- Takahiro Shiota, Characterization of Jacobian varieties in terms of soliton equations, Invent. Math. 83 (1986), no. 2, 333–382. MR 818357, DOI 10.1007/BF01388967
- Christopher Swierczewski and Bernard Deconinck, Computing Riemann theta functions in Sage with applications, Math. Comput. Simulation 127 (2016), 263–272. MR 3501304, DOI 10.1016/j.matcom.2013.04.018
- Marvin D. Tretkoff, The Fermat surface and its periods, Recent developments in several complex variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) Ann. of Math. Stud., vol. 100, Princeton Univ. Press, Princeton, N.J., 1981, pp. 413–428. MR 627771
- C. L. Tretkoff and M. D. Tretkoff, Combinatorial group theory, Riemann surfaces and differential equations, Contributions to group theory, Contemp. Math., vol. 33, Amer. Math. Soc., Providence, RI, 1984, pp. 467–519. MR 767125, DOI 10.1090/conm/033/767125
- Claire Voisin, Hodge theory and complex algebraic geometry. II, Reprint of the 2003 English edition, Cambridge Studies in Advanced Mathematics, vol. 77, Cambridge University Press, Cambridge, 2007. Translated from the French by Leila Schneps. MR 2449178
Additional Information
- Emre Can Sertöz
- Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany
- Email: emresertoz@gmail.com
- Received by editor(s): April 14, 2018
- Received by editor(s) in revised form: December 7, 2018
- Published electronically: April 10, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 2987-3022
- MSC (2010): Primary 32G20, 14C30, 68W30, 14D07, 14K20
- DOI: https://doi.org/10.1090/mcom/3430
- MathSciNet review: 3985484