On a supercongruence conjecture of Rodriguez-Villegas
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Abstract:
In examining the relationship between the number of points over $\mathbb {F}_p$ on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified numerically 22 possible supercongruences. We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the $p$-th Fourier coefficient of a modular form.References
- Scott Ahlgren, Gaussian hypergeometric series and combinatorial congruences, Symbolic computation, number theory, special functions, physics and combinatorics (Gainesville, FL, 1999) Dev. Math., vol. 4, Kluwer Acad. Publ., Dordrecht, 2001, pp. 1–12. MR 1880076, DOI 10.1007/978-1-4613-0257-5_{1}
- Scott Ahlgren and Ken Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math. 518 (2000), 187–212. MR 1739404, DOI 10.1515/crll.2000.004
- 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, DOI 10.1007/BF02101841
- Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams, Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication. MR 1625181
- Jan Stienstra and Frits Beukers, On the Picard-Fuchs equation and the formal Brauer group of certain elliptic $K3$-surfaces, Math. Ann. 271 (1985), no. 2, 269–304. MR 783555, DOI 10.1007/BF01455990
- Jenny G. Fuselier, Hypergeometric functions over $\Bbb F_p$ and relations to elliptic curves and modular forms, Proc. Amer. Math. Soc. 138 (2010), no. 1, 109–123. MR 2550175, DOI 10.1090/S0002-9939-09-10068-0
- John Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), no. 1, 77–101. MR 879564, DOI 10.1090/S0002-9947-1987-0879564-8
- Benedict H. Gross and Neal Koblitz, Gauss sums and the $p$-adic $\Gamma$-function, Ann. of Math. (2) 109 (1979), no. 3, 569–581. MR 534763, DOI 10.2307/1971226
- Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
- Tsuneo Ishikawa, Super congruence for the Apéry numbers, Nagoya Math. J. 118 (1990), 195–202. MR 1060710, DOI 10.1017/S002776300000307X
- Timothy Kilbourn, An extension of the Apéry number supercongruence, Acta Arith. 123 (2006), no. 4, 335–348. MR 2262248, DOI 10.4064/aa123-4-3
- Neal Koblitz, $p$-adic analysis: a short course on recent work, London Mathematical Society Lecture Note Series, vol. 46, Cambridge University Press, Cambridge-New York, 1980. MR 591682, DOI 10.1017/CBO9780511526107
- Neal Koblitz, The number of points on certain families of hypersurfaces over finite fields, Compositio Math. 48 (1983), no. 1, 3–23. MR 700577
- D. McCarthy, $p$-adic hypergeometric series and supercongruences, Ph.D. thesis, University College Dublin, 2010.
- D. McCarthy, Extending Gaussian hypergeometric series to the $p$-adic setting, submitted.
- Dermot McCarthy and Robert Osburn, A $p$-adic analogue of a formula of Ramanujan, Arch. Math. (Basel) 91 (2008), no. 6, 492–504. MR 2465868, DOI 10.1007/s00013-008-2828-0
- Christian Meyer, Modular Calabi-Yau threefolds, Fields Institute Monographs, vol. 22, American Mathematical Society, Providence, RI, 2005. MR 2176545, DOI 10.1090/fim/022
- Eric Mortenson, Supercongruences for truncated $_{n+1}\!F_n$ hypergeometric series with applications to certain weight three newforms, Proc. Amer. Math. Soc. 133 (2005), no. 2, 321–330. MR 2093051, DOI 10.1090/S0002-9939-04-07697-X
- Eric Mortenson, A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory 99 (2003), no. 1, 139–147. MR 1957248, DOI 10.1016/S0022-314X(02)00052-5
- Eric Mortenson, Supercongruences between truncated ${}_2F_1$ hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc. 355 (2003), no. 3, 987–1007. MR 1938742, DOI 10.1090/S0002-9947-02-03172-0
- Fernando Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001) Fields Inst. Commun., vol. 38, Amer. Math. Soc., Providence, RI, 2003, pp. 223–231. MR 2019156
- Chad Schoen, On the geometry of a special determinantal hypersurface associated to the Mumford-Horrocks vector bundle, J. Reine Angew. Math. 364 (1986), 85–111. MR 817640, DOI 10.1515/crll.1986.364.85
- L. Van Hamme, Proof of a conjecture of Beukers on Apéry numbers, Proceedings of the conference on $p$-adic analysis (Houthalen, 1987) Vrije Univ. Brussel, Brussels, 1986, pp. 189–195. MR 921871
Additional Information
- Dermot McCarthy
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 857155
- Email: mccarthy@math.tamu.edu
- Received by editor(s): November 10, 2010
- Received by editor(s) in revised form: February 18, 2011
- Published electronically: November 7, 2011
- Additional Notes: This work was supported by the UCD Ad Astra Research Scholarship program.
- Communicated by: Kathrin Bringmann
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 2241-2254
- MSC (2010): Primary 11F33; Secondary 33C20, 11T24
- DOI: https://doi.org/10.1090/S0002-9939-2011-11087-6
- MathSciNet review: 2898688