AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
On Dwork’s $p$-adic formal congruences theorem and hypergeometric mirror maps
About this Title
E. Delaygue, Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France, T. Rivoal, Institut Fourier, CNRS et Université Grenoble 1, CNRS UMR 5582, 100 rue des maths, BP 74, 38402 St Martin d’Hères cedex, France and J. Roques, Institut Fourier, Université Grenoble 1, CNRS UMR 5582, 100 rue des maths, BP 74, 38402 St Martin d’Hères cedex, France
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 246, Number 1163
ISBNs: 978-1-4704-2300-1 (print); 978-1-4704-3635-3 (online)
DOI: https://doi.org/10.1090/memo/1163
Published electronically: October 13, 2016
Keywords: Dwork’s theory,
generalized hypergeometric functions,
$p$-adic analysis,
integrality of mirror maps
MSC: Primary 11S80; Secondary 14J32, 33C70
Table of Contents
Chapters
- 1. Introduction
- 2. Statements of the main results
- 3. Structure of the paper
- 4. Comments on the main results, comparison with previous results and open questions
- 5. The $p$-adic valuation of Pochhammer symbols
- 6. Proof of Theorem
- 7. Formal congruences
- 8. Proof of Theorem
- 9. Proof of Theorem
- 10. Proof of Theorem
- 11. Proof of Theorem
- 12. Proof of Theorem
- 13. Proof of Corollary
Abstract
Using Dwork’s theory, we prove a broad generalization of his famous $p$-adic formal congruences theorem. This enables us to prove certain $p$-adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number $p$ and not only for almost all primes. Furthermore, using Christol’s functions, we provide an explicit formula for the “Eisenstein constant” of any hypergeometric series with rational parameters.
As an application of these results, we obtain an arithmetic statement “on average” of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It contains all the similar univariate integrality results in the literature, with the exception of certain refinements that hold only in very particular cases.
- G. Almkvist, C. van Enckevort, D. van Straten, and W. Zudilin, Tables of Calabi-Yau equations, arXiv:math/0507430v2 [math.AG] (2010).
- Victor V. Batyrev and Duco van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, Comm. Math. Phys. 168 (1995), no. 3, 493–533. MR 1328251
- Frits Beukers, Algebraic $A$-hypergeometric functions, Invent. Math. 180 (2010), no. 3, 589–610. MR 2609251, DOI 10.1007/s00222-010-0238-y
- F. Beukers and G. Heckman, Monodromy for the hypergeometric function $_nF_{n-1}$, Invent. Math. 95 (1989), no. 2, 325–354. MR 974906, DOI 10.1007/BF01393900
- Michael Bogner and Stefan Reiter, On symplectically rigid local systems of rank four and Calabi-Yau operators, J. Symbolic Comput. 48 (2013), 64–100. MR 2980467, DOI 10.1016/j.jsc.2011.11.007
- Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), no. 1, 21–74. MR 1115626, DOI 10.1016/0550-3213(91)90292-6
- G. Christol, Fonctions hypergéométriques bornées, Groupe de travail d’analyse ultramétrique, tome 14 (1986-1987), exp. 8, 1-16.
- E. Delaygue, Critère pour l’intégralité des coefficients de Taylor des applications miroir, J. Reine Angew. Math. 662 (2012), 205–252 (French, with French summary). MR 2876264, DOI 10.1515/CRELLE.2011.094
- Éric Delaygue, Intégralité des coefficients de Taylor de racines d’applications miroir, J. Théor. Nombres Bordeaux 24 (2012), no. 3, 623–638 (French, with English and French summaries). MR 3010632
- Éric Delaygue, A criterion for the integrality of the Taylor coefficients of mirror maps in several variables, Adv. Math. 234 (2013), 414–452. MR 3003933, DOI 10.1016/j.aim.2012.09.028
- E. Delaygue, Propriétés arithmétiques des applications miroir, PhD thesis (2011), available online at http://math.univ-lyon1.fr/ delaygue/articles/PhD_Thesis_Delaygue.pdf.
- B. Dwork, On $p$-adic differential equations. IV. Generalized hypergeometric functions as $p$-adic analytic functions in one variable, Ann. Sci. École Norm. Sup. (4) 6 (1973), 295–315. MR 572762
- B. Dwork, $p$-adic cycles, Inst. Hautes Études Sci. Publ. Math. 37 (1969), 27–115. MR 294346
- Nadia Heninger, E. M. Rains, and N. J. A. Sloane, On the integrality of $n$th roots of generating functions, J. Combin. Theory Ser. A 113 (2006), no. 8, 1732–1745. MR 2269551, DOI 10.1016/j.jcta.2006.03.018
- Th. Kaluza, Über die Koeffizienten reziproker Potenzreihen, Math. Z. 28 (1928), no. 1, 161–170 (German). MR 1544949, DOI 10.1007/BF01181155
- Nicholas M. Katz, Algebraic solutions of differential equations ($p$-curvature and the Hodge filtration), Invent. Math. 18 (1972), 1–118. MR 337959, DOI 10.1007/BF01389714
- Neal Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, 2nd ed., Graduate Texts in Mathematics, vol. 58, Springer-Verlag, New York, 1984. MR 754003
- Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, Duke Math. J. 151 (2010), no. 2, 175–218. MR 2598376, DOI 10.1215/00127094-2009-063
- Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps. II, Commun. Number Theory Phys. 3 (2009), no. 3, 555–591. MR 2591883, DOI 10.4310/CNTP.2009.v3.n3.a5
- C. Krattenthaler and T. Rivoal, Multivariate $p$-adic formal congruences and integrality of Taylor coefficients of mirror maps, Arithmetic and Galois theories of differential equations, Sémin. Congr., vol. 23, Soc. Math. France, Paris, 2011, pp. 241–269 (English, with English and French summaries). MR 3076084
- C. Krattenthaler and T. Rivoal, Analytic properties of mirror maps, J. Aust. Math. Soc. 92 (2012), no. 2, 195–235. MR 2999156, DOI 10.1017/S1446788712000122
- E. Landau, Sur les conditions de divisibilité d’un produit de factorielles par un autre, collected works, I, page $116$. Thales-Verlag (1985).
- Serge Lang, Cyclotomic fields I and II, 2nd ed., Graduate Texts in Mathematics, vol. 121, Springer-Verlag, New York, 1990. With an appendix by Karl Rubin. MR 1029028
- Bong H. Lian and Shing-Tung Yau, The $n$th root of the mirror map, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001) Fields Inst. Commun., vol. 38, Amer. Math. Soc., Providence, RI, 2003, pp. 195–199. MR 2019153
- Bong H. Lian, Kefeng Liu, and Shing-Tung Yau, Mirror principle. I, Asian J. Math. 1 (1997), no. 4, 729–763. MR 1621573, DOI 10.4310/AJM.1997.v1.n4.a5
- Bong H. Lian and Shing-Tung Yau, Integrality of certain exponential series, Algebra and geometry (Taipei, 1995) Lect. Algebra Geom., vol. 2, Int. Press, Cambridge, MA, 1998, pp. 215–227. MR 1697956
- Anton Mellit and Masha Vlasenko, Dwork’s congruences for the constant terms of powers of a Laurent polynomial, Int. J. Number Theory 12 (2016), no. 2, 313–321. MR 3461433, DOI 10.1142/S1793042116500184
- Yasuo Morita, A $p$-adic analogue of the $\Gamma$-function, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 2, 255–266. MR 424762
- Kaori Ota, On special values of generalized $p$-adic hypergeometric functions, Acta Arith. 67 (1994), no. 2, 141–163. MR 1291873, DOI 10.4064/aa-67-2-141-163
- Alain M. Robert, A course in $p$-adic analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000. MR 1760253
- Julien Roques, Arithmetic properties of mirror maps associated with Gauss hypergeometric equations, Monatsh. Math. 171 (2013), no. 2, 241–253. MR 3077934, DOI 10.1007/s00605-013-0505-2
- Julien Roques, On generalized hypergeometric equations and mirror maps, Proc. Amer. Math. Soc. 142 (2014), no. 9, 3153–3167. MR 3223372, DOI 10.1090/S0002-9939-2014-12161-7
- Jan Stienstra, GKZ hypergeometric structures, Arithmetic and geometry around hypergeometric functions, Progr. Math., vol. 260, Birkhäuser, Basel, 2007, pp. 313–371. MR 2306158, DOI 10.1007/978-3-7643-8284-1_{1}2
- V. V. Zudilin, On the integrality of power expansions related to hypergeometric series, Mat. Zametki 71 (2002), no. 5, 662–676 (Russian, with Russian summary); English transl., Math. Notes 71 (2002), no. 5-6, 604–616. MR 1936191, DOI 10.1023/A:1015827602930