Supercongruences for truncated _{𝑛+1}𝐹_{𝑛} hypergeometric series with applications to certain weight three newforms

Mortenson, Eric

doi:10.1090/S0002-9939-04-07697-X

Supercongruences for truncated ${}_{n+1}F_{n}$ hypergeometric series with applications to certain weight three newforms
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Proc. Amer. Math. Soc. 133 (2005), 321-330 Request permission

Abstract:

We prove general results on supercongruences between values of truncated $_{n+1}F_{n}$ hypergeometric functions and their character analogs. As a consequence of the main results of this paper, we prove Beukers-type supercongruences for certain weight three newforms.

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
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
Bruce C. Berndt and Ronald J. Evans, Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer, Illinois J. Math. 23 (1979), no. 3, 374–437. MR 537798
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
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
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
Ken Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc. 350 (1998), no. 3, 1205–1223. MR 1407498, DOI 10.1090/S0002-9947-98-01887-X
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
F. Rodriguez-Villegas, private communication.
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
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