×

zbMATH — the first resource for mathematics

A numerical transcendental method in algebraic geometry: computation of Picard groups and related invariants. (English) Zbl 1441.14036

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14C22 Picard groups
32G20 Period matrices, variation of Hodge structure; degenerations
14Q10 Computational aspects of algebraic surfaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Elliptic Curves, Modular Forms and Cryptography, in Proceedings of the Advanced Instructional Workshop on Algebraic Number Theory, A. K. Bhandari, D. S. Nagaraj, B. Ramakrishnan and T. N. Venkataramana, eds., Hindustan Book Agency, New Delhi, 2003.
[2] T. G. Abbott, K. S. Kedlaya, and D. Roe, Bounding Picard numbers of surfaces using \(p\)-adic cohomology, in Arithmetics, Geometry, and Coding Theory (AGCT 2005), Sémin. Congr. 21, Soc. Math. France, Paris, 2010, pp. 125–-159. · Zbl 1214.14007
[3] N. Addington and A. Auel, Some Non-special Cubic Fourfolds, preprint, https://arxiv.org/abs/1703.05923, 2018.
[4] V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps, Volume 2, Monogr. Math. 83, Birkhäuser Boston, Boston, 1988.
[5] D. Bailey, Integer relation detection, Comput. Sci. Eng., 2 (2008), pp. 24–48, https://doi.org/10.1109/5992.814653.
[6] W. Barth, K. Hulek, C. Peters, and A. van de Ven, Compact Complex Surfaces, A Series of Modern Surveys in Mathematics, 2nd ed., Springer-Verlag, Berlin, Heidelberg, 2004.
[7] A. R. Booker, J. Sijsling, A. V. Sutherland, J. Voight, and D. Yasaki, A database of genus-\(2\) curves over the rational numbers, LMS J. Comput. Math., 19 (2016), pp. 235–254, https://doi.org/10.1112/S146115701600019X. · Zbl 1404.11090
[8] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), pp. 235–265, https://doi.org/10.1006/jsco.1996.0125. · Zbl 0898.68039
[9] A. Bostan, P. Lairez, and B. Salvy, Creative telescoping for rational functions using the Griffiths–Dwork method, in Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2013, pp. 93–100, https://doi.org/10.1145/2465506.2465935. · Zbl 1360.68921
[10] D. J. Broadhurst and D. Kreimer, Knots and numbers in \(\phi^4\) theory to 7 loops and beyond, Internat. J. Modern Phys. C, 6 (1995), pp. 519–-524, https://doi.org/10.1142/S012918319500037X. · Zbl 0940.81520
[11] F. Brown, Mixed Tate motives over \(\mathbb{Z} \), Ann. of Math. (2), 175 (2012), pp. 949-–976, https://doi.org/10.4007/annals.2012.175.2.10. · Zbl 1278.19008
[12] N. Bruin, J. Sijsling, and A. Zotine, Numerical computation of endomorphism rings of Jacobians, Open Book Ser., 2 (2019), pp. 155–171, https://doi.org/10.2140/obs.2019.2.155.
[13] J. Buchmann and M. Pohst, Computing a lattice basis from a system of generating vectors, in Eurocal ’87, J. H. Davenport, ed., Lecture Notes in Comput. Sci. 378, Springer, Berlin, Heidelberg, 1989, pp. 54–63. · Zbl 1209.11108
[14] F. Charles, On the Picard number of K3 surfaces over number fields, Algebra Number Theory, 8 (2014), pp. 1–17, https://doi.org/10.2140/ant.2014.8.1. · Zbl 1316.14069
[15] J. Chen, D. Stehlé, and G. Villard, A new view on HJLS and PSLQ: Sums and projections of lattices, in Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, ISSAC ’13, ACM, New York, 2013, pp. 149–156, https://doi.org/10.1145/2465506.2465936. · Zbl 1360.11141
[16] D. V. Chudnovsky and G. V. Chudnovsky, Computer algebra in the service of mathematical physics and number theory, in Computers in Mathematics (Stanford, CA, 1986), Lecture Notes in Pure and Appl. Math. 125, Dekker, New York, 1990, pp. 109–232.
[17] F. Chyzak, An extension of Zeilberger’s fast algorithm to general holonomic functions, Discrete Math., 217 (2000), pp. 115–134, https://doi.org/10.1016/S0012-365X(99)00259-9. · Zbl 0968.33011
[18] C. Ciliberto, J. Harris, and R. Miranda, General components of the Noether-Lefschetz locus and their density in the space of all surfaces, Math. Ann., 282 (1988), pp. 667–680. · Zbl 0671.14017
[19] E. Costa, Effective Computations of Hasse–Weil Zeta Functions, ProQuest LLC, Ann Arbor, MI, Ph.D. Thesis, New York University, New York, 2015.
[20] E. Costa, D. Harvey, and K. S. Kedlaya, Zeta Functions of Nondegenerate Hypersurfaces in Toric Varieties via cOntrolled Reduction in \(p\)-adic Cohomology, preprint, https://arxiv.org/abs/1806.00368, 2018.
[21] E. Costa, N. Mascot, J. Sijsling, and J. Voight, Rigorous computation of the endomorphism ring of a Jacobian, Math. Comp., 88 (2019), pp. 1303–1339. · Zbl 07009723
[22] B. Deconinck and M. S. Patterson, Computing with plane algebraic curves and Riemann surfaces: The algorithms of the Maple package “algcurves”, Lecture Notes in Math. 2013, Springer, Heidelberg, 2011, pp. 67–123. · Zbl 1213.14114
[23] A. Degtyarev and I. Shimada, On the topology of projective subspaces in complex Fermat varieties, J. Math. Soc. Japan, 68 (2016), pp. 975–996. · Zbl 1354.14032
[24] P. Deligne, The Hodge conjecture, in The Millennium Prize Problems, Clay Math. Inst., Cambridge, MA, 2006, pp. 45–53.
[25] S. Di Rocco, D. Eklund, C. Peterson, and A. J. Sommese, Chern numbers of smooth varieties via homotopy continuation and intersection theory, J. Symbolic Comput., 46 (2011), pp. 23–33, https://doi.org/10.1016/j.jsc.2010.06.026. · Zbl 1200.14014
[26] I. V. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces, J. Math. Sci., 81 (1996), pp. 2599–2630, https://doi.org/10.1007/BF02362332. · Zbl 0890.14024
[27] I. V. Dolgachev, Luigi Cremona and cubic surfaces, in Luigi Cremona (1830–1903) (Italian), Incontr. Studio 36, Istituto Lombardo di Scienze e Lettere, Milan, 2005, pp. 55–70.
[28] A.-S. Elsenhans and J. Jahnel, On the computation of the Picard group for \(K 3\) surfaces, Math. Proc. Camb. Philos. Soc., 151 (2011), pp. 263–270, https://doi.org/10.1017/S0305004111000326. · Zbl 1223.14044
[29] A.-S. Elsenhans and J. Jahnel, The picard group of a \(K 3\) surface and its reduction modulo \(p\), Algebra Number Theory, 5 (2011), pp. 1027–1040, https://doi.org/10.2140/ant.2011.5.1027. · Zbl 1243.14014
[30] A.-S. Elsenhans and J. Jahnel, Kummer surfaces and the computation of the Picard group, LMS J. Comput. Math., 15 (2012), pp. 84–100. · Zbl 1297.14043
[31] Y. Feng, J. Chen, and W. Wu, The PSLQ algorithm for empirical data, Math. Comp., 88 (2019), pp. 1479–1501, https://doi.org/10.1090/mcom/3356. · Zbl 07009728
[32] H. R. P. Ferguson, D. H. Bailey, and S. Arno, Analysis of PSLQ, an integer relation finding algorithm, Math. Comput., 68 (1999), pp. 351–370, https://doi.org/10.1090/S0025-5718-99-00995-3. · Zbl 0927.11055
[33] H. R. P. Ferguson and R. W. Forcade, Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two, Bull. Am. Math. Soc., 1 (1979), pp. 912–914, https://doi.org/10.1090/S0273-0979-1979-14691-3. · Zbl 0424.10021
[34] D. Festi, A Practical Algorithm to Compute the Geometric Picard Lattice of \(K 3\) Surfaces of Degree \(2\), preprint, https://arxiv.org/abs/1808.00351, 2018.
[35] D. Festi and D. van Straten, Bhabha Scattering and a Special Pencil of \(K 3\) Surfaces, preprint, https://arxiv.org/abs/1809.04970, 2018.
[36] U. Fincke and M. Pohst, A procedure for determining algebraic integers of given norm, in Computer Algebra, J. A. van Hulzen, ed., Lecture Notes in Comput. Sci. 162, Springer, Berlin, 1983, pp. 194–202. · Zbl 0541.12001
[37] F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions. Vol. I, Cambridge Stud. Adv. Math. 79, Cambridge University Press, Cambridge, 2003. · Zbl 1061.37056
[38] P. A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2), 90 (1969), pp. 460–495; ibid. (2), 90 (1969), pp. 496–541.
[39] P. A. Griffiths, A transcendental method in algebraic geometry, in Actes Du Congrès International Des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, pp. 113–119.
[40] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York, 1977.
[41] B. Hassett, A. Kresch, and Y. Tschinkel, Effective computation of Picard groups and Brauer-Manin obstructions of degree two \(K 3\) surfaces over number fields, Rendiconti Circolo Mat. Palermo, 62 (2013), pp. 137–151, https://doi.org/10.1007/s12215-013-0116-8. · Zbl 1297.14027
[42] J. Hastad, B. Just, J. C. Lagarias, and C. P. Schnorr, Polynomial time algorithms for finding integer relations among real numbers, SIAM J. Comput., 18 (1989), pp. 859–881, https://doi.org/10.1137/0218059. · Zbl 0692.10033
[43] J. D. Hauenstein, J. I. Rodriguez, and F. Sottile, Numerical computation of Galois groups, Found Comput Math, 18 (2018), pp. 867–890, https://doi.org/10.1007/s10208-017-9356-x. · Zbl 1442.14186
[44] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror Symmetry, Clay Math. Monogr. 1, AMS, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2003.
[45] C. Houzel, La Géométrie Algébrique, Librairie Scientifique et Technique, Albert Blanchard, Paris, 2002.
[46] D. Huybrechts, Lectures on K3 Surfaces, Cambridge Stud. Adv. Math. 158, Cambridge University Press, Cambridge, UK, 2016.
[47] R. Kannan, Improved algorithms for integer programming and related lattice problems, in Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, STOC ’83,ACM, New York, 1983, pp. 193–206, https://doi.org/10.1145/800061.808749.
[48] K. S. Kedlaya, Computing zeta functions via \(p\)-adic cohomology, in Algorithmic Number Theory, Lecture Notes in Comput. Sci. 3076, Springer, Berlin, 2004, pp. 1–17. · Zbl 1125.14300
[49] C. Koutschan, A fast approach to creative telescoping, Math. Comput. Sci., 4 (2010), pp. 259–266, https://doi.org/10.1007/s11786-010-0055-0. · Zbl 1218.68205
[50] P. Lairez, Computing periods of rational integrals, Math. Comp., 85 (2016), pp. 1719–1752, https://doi.org/10.1090/mcom/3054. · Zbl 1337.68301
[51] A. G. B. Lauder, Counting solutions to equations in many variables over finite fields, Found. Comput. Math., 4 (2004), pp. 221–267.
[52] S. Lefschetz, L’analysis situs et la géométrie algébrique, Gauthier-Villars, Paris, 1950. · Zbl 0035.10202
[53] A. K. Lenstra, H. W. Lenstra, and L. Lovàsz, Factoring polynomials with rational coefficients, Math. Ann., 261 (1982), pp. 515–534, https://doi.org/10.1007/BF01457454.
[54] D. Lombardo, Computing the geometric endomorphism ring of a genus-\(2\) Jacobian, Math. Comp., 88 (2019), pp. 889–929. · Zbl 1410.11043
[55] E. Looijenga, Fermat varieties and the periods of some hypersurfaces, in Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo 2007), Adv. Stud. Pure Math. 58, Math. Soc. Japan, Tokyo, 2010, pp. 47–67.
[56] M. Mezzarobba, NumGFun: A package for numerical and analytic computation with D-finite functions, in Proceedings of the 35th International Symposium on Symbolic and Algebraic Computation, S. M. Watt, ed., Munich, Germany, ACM, New York, 2010, pp. 139–145, https://doi.org/10.1145/1837934.1837965. · Zbl 1321.65202
[57] M. Mezzarobba, Rigorous Multiple-precision Evaluation of D-finite Functions in SageMath, preprint, https://arxiv.org/abs/1607.01967, 2016.
[58] H. Movasati, A Course in Hodge Theory, with Emphasis in Multiple Integrals, to appear, https://w3.impa.br/ hossein/myarticles/hodgetheory.pdf.
[59] H. Movasati, Calculation of mixed Hodge structures, Gauss-Manin connections and Picard-Fuchs equations, in Real and Complex Singularities, Trends Math., Birkhäuser, Basel, 2007, pp. 247–262. · Zbl 1117.14014
[60] P. Q. Nguyen, Hermite’s constant and lattice algorithms, in The LLL Algorithm, P. Q. Nguyen and B. Vallée, eds., Springer-Verlag, Berlin Heidelberg, 2009, pp. 19–69, https://doi.org/10.1007/978-3-642-02295-1_2.
[61] S. Pancratz, Computing Gauss–Manin Connections for Families of Projectives Hypersurfaces, 2010, http://www.pancratz.org/files/Transfer20100309.pdf.
[62] S. Pancratz and J. Tuitman, Improvements to the deformation method for counting points on smooth projective hypersurfaces, Found. Comput. Math., 15 (2015), pp. 1413–1464. · Zbl 1400.11128
[63] F. Pham, Formules de Picard-Lefschetz généralisées et ramification des intégrales, Bull. Soc. Math. France, 93 (1965), pp. 333–367. · Zbl 0192.29701
[64] E. Picard, Sur les périodes des intégrales doubles et sur une classe d’équations différentielles linéaires, Ann. Sci. École Norm. Sup. (3), 50 (1933), pp. 393–395. · Zbl 0008.15801
[65] B. Poonen, D. Testa, and R. van Luijk, Computing Néron–Severi groups and cycle class groups, Compos. Math., 151 (2015), pp. 713–734, https://doi.org/10.1112/S0010437X14007878. · Zbl 1316.14017
[66] K. Ranestad and C. Voisin, Variety of power sums and divisors in the moduli space of cubic fourfolds, Doc. Math., 22 (2017), pp. 455–504. · Zbl 1369.14056
[67] W. M. Schmidt, Linear forms with algebraic coefficients. I, J. Number Theory, 3 (1971), pp. 253–277, https://doi.org/10.1016/0022-314X(71)90001-1.
[68] E. C. Sertöz, Computing periods of hypersurfaces, Math. Comp., 88 (2019), pp. 2987–3022, https://doi.org/10.1090/mcom/3430.
[69] T. Shioda, An explicit algorithm for computing the Picard number of certain algebraic surfaces, Amer. J. Math., 108 (1986), pp. 415–432. · Zbl 0602.14033
[70] T. Shiota, Characterization of Jacobian varieties in terms of soliton equations, Invent. Math., 83 (1986), pp. 333–382. · Zbl 0621.35097
[71] A. J. Sommese, J. Verschelde, and C. W. Wampler, Numerical decomposition of the solution sets of polynomial systems into irreducible components, SIAM J. Numer. Anal., 38 (2001), pp. 2022–2046, https://doi.org/10.1137/S0036142900372549. · Zbl 1002.65060
[72] The LMFDB Collaboration, The l-Functions and Modular Forms Database, http://www.lmfdb.org, 2013, accessed September 16, 2013.
[73] The Sage Developers, SageMath, the Sage Mathematics Software System (Version 8.4), 2018, https://wiki.sagemath.org/Publications_using_SageMath?action=show&redirect=Publications_using_SAGE.
[74] I. Todorov, Number theory meets high energy physics, Phys. Part. Nuclei Lett., 14 (2017), pp. 291–297, https://doi.org/10.1134/S1547477117020339.
[75] P. Tretkoff, Periods and Special Functions in Transcendence, Advanced Textbooks in Mathematics, World Scientific, Hackensack, NJ, 2017.
[76] J. van der Hoeven, Fast evaluation of holonomic functions near and in regular singularities, J. Symbolic Comput., 31 (2001), pp. 717–743, https://doi.org/10.1006/jsco.2000.0474. · Zbl 0982.65024
[77] J. van der Hoeven, On effective analytic continuation, Math. Comput. Sci., 1 (2007), pp. 111–175, https://doi.org/10.1007/s11786-007-0006-6.
[78] R. van Luijk, K3 surfaces with Picard number one and infinitely many rational points, Algebra Number Theory, 1 (2007), pp. 1–15, https://doi.org/10.2140/ant.2007.1.1. · Zbl 1123.14022
[79] P. van Wamelen, Examples of genus two CM curves defined over the rationals, Math. Comp., 68 (1999), pp. 307–320, https://doi.org/10.1090/S0025-5718-99-01020-0. · Zbl 0906.14025
[80] P. Vanhove, The physics and the mixed Hodge structure of Feynman integrals, in Proceedings of Symposia in Pure Mathematics, 88 R. Donagi, M. Douglas, L. Kamenova, and M. Rocek, eds., AMS, Providence, RI, 2014, pp. 161–194, https://doi.org/10.1090/pspum/088/01455.
[81] C. Voisin, The Hodge Conjecture, in Open Problems in Mathematics, Springer, Cham, 2016, pp. 521–543.
[82] Y. G. Zarhin, Hodge groups of K3 surfaces, J. Reine Angew. Math., 341 (1983), pp. 193–220.
[83] O. Zariski, Algebraic Surfaces, Classics in Mathematics, Springer-Verlag, Berlin, 1995, reprint of the second (1971) edition.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.