Abstract
Mirror maps are power series which occur in Mirror Symmetry as the inverse for composition of \(q(z)=\exp (f(z)/g(z))\), called local q-coordinates, where f and g are particular solutions of the Picard–Fuchs differential equations associated with certain one-parameter families of Calabi–Yau varieties. In several cases, it has been observed that such power series have integral Taylor coefficients at the origin. In the case of hypergeometric equations, we discuss p-adic tools and techniques that enable one to prove a criterion for the integrality of the coefficients of mirror maps. This is a joint work with T. Rivoal and J. Roques. This note is an extended abstract of the talk given by the author in January 2017 at the conference “Hypergeometric motives and Calabi–Yau differential equations” in Creswick, Australia.
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References
Apéry, R.: Irrationalité de ζ(2) et ζ(3). Astérisque 61, 11–13 (1979)
Bober, J.W.: Factorial ratios, hypergeometric series, and a family of step functions. J. Lond. Math. Soc. (2) 79, 422–444 (2009)
Christol, G.: Fonctions hypergéométriques bornées. Groupe de travail d’analyse ultramétrique, tome 14, exp. 8, 1–16 (1986–1987)
Delaygue, É.: Critère pour l’intégralité des coefficients de Taylor des applications miroir. J. Reine Angew. Math. 662, 205–252 (2012)
Delaygue, É.: A criterion for the integrality of the Taylor coefficients of mirror maps in several variables. Adv. Math. 234, 414–452 (2013)
Delaygue, É., Rivoal, T., Roques, J.: On Dwork’s p-adic formal congruences theorem and hypergeometric mirror maps. Mem. Am. Math. Soc. 246(1163), 100 pp (2017)
Dwork, B.: On p-adic differential equations IV generalized hypergeometric functions as p-adic analytic functions in one variable. Annales Scientifiques de l’É.N.S. 4o série, tome 6(3), 295–316 (1973)
Krattenthaler, C., Rivoal, T.: On the integrality of the Taylor coefficients of mirror maps. Duke Math. J. 151, 175–218 (2010)
Landau, E.: Sur les conditions de divisibilité d’un produit de factorielles par un autre. Collected works, I, p. 116. Thales-Verlag, New Cairo (1985)
Lian, B.H., Yau, S.T.: Integrality of certain exponential series. Algebra and Geometry (Taipei, 1995), pp. 215–227. Lectures on Algebraic Geometry, vol. 2. International Press, Cambridge (1998)
Zudilin, V.V.: Integrality of power expansions related to hypergeometric series. Math. Notes 71(5), 604–616 (2002)
Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under the Grant Agreement No 648132.
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Delaygue, É. (2019). Arithmetic Properties of Hypergeometric Mirror Maps and Dwork’s Congruences. In: de Gier, J., Praeger, C., Tao, T. (eds) 2017 MATRIX Annals. MATRIX Book Series, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-030-04161-8_28
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DOI: https://doi.org/10.1007/978-3-030-04161-8_28
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