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ANALYTIC PROPERTIES OF MIRROR MAPS

Part of: Series expansions Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  22 November 2012

C. KRATTENTHALER  and
T. RIVOAL
C. KRATTENTHALER
Affiliation:
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Vienna, Austria (email: Christian.Krattenthaler@univie.ac.at)
T. RIVOAL*
Affiliation:
Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères cedex, France
*
✉For correspondence; e-mail: rivoal@math.univ-lyon1.fr
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Abstract

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We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin. This family includes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated ‘quintic’ example of Candelas, de la Ossa, Green and Parkes. In a previous paper, we proved that all coefficients in the Taylor expansions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in their Taylor expansions at 0 are positive. Furthermore, we provide several results about the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address their analytic continuation, points of singularity, and radius of covergence. We present several very precise conjectures on the radius of covergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0.

Keywords

mirror mapscanonical coordinatesanalytic continuationsingular expansiongeneralised hypergeometric functionsmodular forms

MSC classification

Secondary: 30B10: Power series (including lacunary series) 14J32: Calabi-Yau manifolds 30B40: Analytic continuation 33C20: Generalized hypergeometric series, $_pF_q$
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 2 , April 2012 , pp. 195 - 235
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

The research of the first author was partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607-N13, the latter in the framework of the National Research Network ‘Analytic Combinatorics and Probabilistic Number Theory’. The research of the second author was partially supported by the project Q-DIFF, ANR-10-JCJC-0105, of the French ‘Agence Nationale de la Recherche’.

References

[1]Almkvist, G. and Zudilin, W., ‘Differential equations, mirror maps and zeta values’, in: Mirror Symmetry V, AMS/IP Studies in Advanced Mathematics, 38 (eds. Yui, N., Yau, S.-T. and Lewis, J. D.) (American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2007), pp. 481–515.Google Scholar
[2]André, Y., ‘G-fonctions et transcendance’, J. reine angew. Math. 476 (1996), 95–125.Google Scholar
[3]Andrews, G. E., Askey, R. A. and Roy, R., Special Functions, The Encyclopedia of Mathematics and Its Applications, 71 (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
[4]Asai, T., Kaneko, M. and Ninomiya, H., ‘Zeros of certain modular functions and an application’, Comment. Math. Univ. St. Pauli 46 (1997), 93–101.Google Scholar
[5]Bach, E. and Shallit, J. O., Algorithmic Number Theory, Vol. 1 (MIT Press, Cambridge, MA, 1996).Google Scholar
[6]Bailey, W. N., Generalized Hypergeometric Series (Cambridge University Press, Cambridge, 1935).Google Scholar
[7]Beukers, F., ‘Irrationality proofs using modular forms’, in: Journées arithmétiques de Besançon (Besançon, 1985), Astérisque, 147–148 (Soc. Math. France, Paris, 1987), pp. 271–283, 345.Google Scholar
[8]Beukers, F. and Heckman, G., ‘Monodromy for the hypergeometric function nF n−1’, Invent. Math. 95 (1989), 325–354.CrossRefGoogle Scholar
[9]Borwein, J. M. and Borwein, P. B., ‘A cubic counterpart of Jacobi’s identity and the AGM’, Trans. Amer. Math. Soc. 323 (1991), 691–701.Google Scholar
[10]Candelas, P., de la Ossa, X., Green, P. and Parkes, L., ‘A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory’, Nucl. Phys. B359 (1991), 21–74.CrossRefGoogle Scholar
[11]Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).Google Scholar
[12]De Bruijn, N. G., Asymptotic Methods in Analysis (North-Holland, Amsterdam, 1958).Google Scholar
[13]Delaygue, E., ‘Critère pour l’integralité des coefficients de Taylor des applications miroir’, J. reine angew. Math. 662 (2012), 205–252.Google Scholar
[14]Flajolet, P. and Sedgewick, R., Analytic Combinatorics (Cambridge University Press, Cambridge, 2009).CrossRefGoogle Scholar
[15]Garvan, F., ‘Shifted and shiftless partition identities II’, in: Number Theory for the Millennium II, (eds. Bennett, M. A., Berndt, B. C., Boston, N., Diamond, H. G., Hildebrand, A. J. and Philipp, W.) (A K Peters, Natick, MA, 2002), pp. 75–92.Google Scholar
[16]Hausdorff, F., ‘Momentprobleme für ein endliches Intervall’, Math. Z. 16 (1923), 220–248.CrossRefGoogle Scholar
[17]Henrici, P., Applied and Computational Complex Analysis, Vol. 1 (Wiley, New York, 1974).Google Scholar
[18]Herrmann, O., ‘Über die Berechnung der Fourierkoeffizienten der Funktion j(τ)’, J. reine angew. Math. 274/275 (1973/74), 187–195.Google Scholar
[19]Kaluza, T., ‘Über die Koeffizienten reziproker Potenzreihen’, Math. Z. 28 (1928), 161–170.CrossRefGoogle Scholar
[20]Krattenthaler, C. and Rivoal, T., ‘On the integrality of Taylor coefficients of mirror maps’, Duke Math. J. 151 (2010), 175–218.CrossRefGoogle Scholar
[21]Krattenthaler, C. and Rivoal, T., ‘Multivariate p-adic formal congruences and integrality of Taylor coefficients of mirror maps’, in: Théories Galoisiennes et Arithmétiques des Équations Différentielles, Séminaires et Congrès, 27 (eds. Di Vizio, L. and Rivoal, T.) (Société Mathématique de France, Paris, 2011), pp. 279–307.Google Scholar
[22]Lamperti, J., ‘On the coefficients of reciprocal power series’, Amer. Math. Monthly 65 (1958), 90–94.CrossRefGoogle Scholar
[23]Lau, S.-C., Leung, N. C. and Wu, B., ‘Mirror maps equal SYZ maps for toric Calabi–Yau surfaces’, Bull. London Math. Soc. 44 (2012), 255–270.CrossRefGoogle Scholar
[24]Lian, B. H. and Yau, S.-T., ‘Arithmetic properties of mirror map and quantum coupling’, Comm. Math. Phys. 176 (1996), 163–191.CrossRefGoogle Scholar
[25]Lian, B. H. and Yau, S.-T., ‘Mirror maps, modular relations and hypergeometric series I’, appeared as ‘Integrality of certain exponential series’, in: Lectures in Algebra and Geometry, Proceedings of the International Conference on Algebra and Geometry (Taipei, 1995) (ed. Kang, M.-C.) (International Press, Cambridge, MA, 1998), pp. 215–227.Google Scholar
[26]Morrison, D. R., ‘Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians’, J. Amer. Math. Soc. 6 (1993), 223–247.CrossRefGoogle Scholar
[27]Pandharipande, R., ‘Rational curves on hypersurfaces (after A. Givental)’, in: Séminaire Bourbaki, Vol. 1997/98, Astérisque, 252 (Soc. Math. France, Paris, 1998), Exp. No. 848, 5, 307–340.Google Scholar
[28] The PARI Group, PARI/GP. http://pari.math.u-bordeaux.fr.Google Scholar
[29]Pólya, G., ‘Application of a theorem connected with the problem of moments’, Mess. Math. 55 (1926), 189–192.Google Scholar
[30]van der Poorten, A. J., ‘A proof that Euler missed: Apéry’s proof of the irrationality of ζ(3)’, Math. Intelligencer 1 (1978/79), 195–203.CrossRefGoogle Scholar
[31]Rankin, R. A., Modular Forms and Functions (Cambridge University Press, Cambridge, 1977).CrossRefGoogle Scholar
[32]Sasvári, Z., ‘An elementary proof of Binet’s formula for the gamma function’, Amer. Math. Monthly 106 (1999), 156–158.CrossRefGoogle Scholar
[33]Serre, J.-P., Cours d’arithmétique (deuxième édition revue et corrigée), Le Mathématicien, 2, (Presses Universitaires de France, Paris, 1977).Google Scholar
[34]Slater, L. J., Generalized Hypergeometric Functions (Cambridge University Press, Cambridge, 1966).Google Scholar
[35]Stiller, P. F., ‘Classical automorphic forms and hypergeometric functions’, J. Number Theory 28 (1988), 219–232.CrossRefGoogle Scholar
[36]Voisin, C., Mirror Symmetry, SMF/AMS Texts and Monographs, 1 (American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 1999).Google Scholar
[37]Wirths, K. J., ‘Über totalmonotone Zahlenfolgen’, Arch. Math. 26 (1975), 508–517.CrossRefGoogle Scholar
[38]Zudilin, W., ‘Number theory casting a look at the mirror’, Preprint, arXiv:math.NT/0008237.Google Scholar
[39]Zudilin, W., ‘Integrality of power expansions related to hypergeometric series’, Math. Notes 71 (2002), 604–616.CrossRefGoogle Scholar
[40]Zudilin, W., ‘The hypergeometric equation and Ramanujan functions’, Ramanujan J. 7 (2003), 435–447.CrossRefGoogle Scholar