Supercongruences between truncated $_{2}F_{1}$ hypergeometric functions and their Gaussian analogs
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- by Eric Mortenson
- Trans. Amer. Math. Soc. 355 (2003), 987-1007
- DOI: https://doi.org/10.1090/S0002-9947-02-03172-0
- Published electronically: October 25, 2002
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Abstract:
Fernando Rodriguez-Villegas has conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension $d\le 3$. For manifolds of dimension $d=1$, he observed four potential supercongruences. Later the author proved one of the four. Motivated by Rodriguez-Villegas’s work, in the present paper we prove a general result on supercongruences between values of truncated $_{2}F_{1}$ hypergeometric functions and Gaussian hypergeometric functions. As a corollary to that result, we prove the three remaining supercongruences.References
- S. Ahlgren, Gaussian hypergeometric series and combinatorial congruences, Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Dev. Math., 4, Kluwer, Dordrecht, 2001, pp. 1–12.
- 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
- F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201–210. MR 873877, DOI 10.1016/0022-314X(87)90025-4
- P. Candelas, X. de la Ossa, and F. Rodriguez-Villegas, Calabi-Yau manifolds over finite fields I, http://xxx.lanl.gov/abs/hep-th/0012233.
- 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
- Tsuneo Ishikawa, On Beukers’ conjecture, Kobe J. Math. 6 (1989), no. 1, 49–51. MR 1023525
- Kenneth F. Ireland and Michael I. Rosen, A classical introduction to modern number theory, Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York-Berlin, 1982. Revised edition of Elements of number theory. MR 661047
- E. Mortenson, A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory, to appear.
- Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger, $A=B$, A K Peters, Ltd., Wellesley, MA, 1996. With a foreword by Donald E. Knuth; With a separately available computer disk. MR 1379802
- F. Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, preprint.
- F. Rodriguez-Villegas, private communication.
Bibliographic Information
- Eric Mortenson
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- Email: mort@math.wisc.edu
- Received by editor(s): February 27, 2002
- Received by editor(s) in revised form: July 22, 2002
- Published electronically: October 25, 2002
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 987-1007
- MSC (2000): Primary 11F85, 11L10
- DOI: https://doi.org/10.1090/S0002-9947-02-03172-0
- MathSciNet review: 1938742