## Hypergeometric type identities in the $$p$$-adic setting and modular forms.(English)Zbl 1397.11078

Summary: We prove hypergeometric type identities for a function defined in terms of quotients of the $$p$$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $$_4F_3$$ hypergeometric series and the Fourier coefficients of a certain weight four modular form.

### MSC:

 11F33 Congruences for modular and $$p$$-adic modular forms 33C20 Generalized hypergeometric series, $${}_pF_q$$ 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 33E50 Special functions in characteristic $$p$$ (gamma functions, etc.)

### Keywords:

hypergeometric identities; modular forms

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### References:

 [1] Ahlgren, Scott; Ono, Ken, A Gaussian hypergeometric series evaluation and Ap\'ery number congruences, J. Reine Angew. Math., 518, 187-212 (2000) · Zbl 0940.33002 [2] Andrews, George E.; Askey, Richard; Roy, Ranjan, Special functions, Encyclopedia of Mathematics and its Applications 71, xvi+664 pp. (1999), Cambridge University Press, Cambridge · Zbl 0920.33001 [3] Berndt, Bruce C.; Evans, Ronald J.; Williams, Kenneth S., Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, xii+583 pp. (1998), John Wiley & Sons, Inc., New York · Zbl 0906.11001 [4] Barman, Rupam; Saikia, Neelam, $$p$$-adic gamma function and the trace of Frobenius of elliptic curves, J. Number Theory, 140, 181-195 (2014) · Zbl 1345.11041 [5] Barman, Rupam; Saikia, Neelam, Certain transformations for hypergeometric series in the $$p$$-adic setting, Int. J. Number Theory, 11, 2, 645-660 (2015) · Zbl 1365.11121 [6] Evans, Ron, Hypergeometric $$_3F_2(1/4)$$ evaluations over finite fields and Hecke eigenforms, Proc. Amer. Math. Soc., 138, 2, 517-531 (2010) · Zbl 1268.11158 [7] Fuselier, Jenny G., Hypergeometric functions over finite fields and relations to modular forms and elliptic curves, 63 pp. (2007), ProQuest LLC, Ann Arbor, MI [8] Fuselier, Jenny G., Hypergeometric functions over $$\mathbb{F}_p$$ and relations to elliptic curves and modular forms, Proc. Amer. Math. Soc., 138, 1, 109-123 (2010) · Zbl 1222.11058 [9] Fuselier, Jenny G., Traces of Hecke operators in level 1 and Gaussian hypergeometric functions, Proc. Amer. Math. Soc., 141, 6, 1871-1881 (2013) · Zbl 1297.11026 [10] Frechette, Sharon; Ono, Ken; Papanikolas, Matthew, Gaussian hypergeometric functions and traces of Hecke operators, Int. Math. Res. Not., 60, 3233-3262 (2004) · Zbl 1088.11029 [11] Greene, John, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc., 301, 1, 77-101 (1987) · Zbl 0629.12017 [12] Gross, Benedict H.; Koblitz, Neal, Gauss sums and the $$p$$-adic $$\Gamma$$-function, Ann. of Math. (2), 109, 3, 569-581 (1979) · Zbl 0406.12010 [13] Katz, Nicholas M., Exponential sums and differential equations, Annals of Mathematics Studies 124, xii+430 pp. (1990), Princeton University Press, Princeton, NJ · Zbl 0731.14008 [14] Kilbourn, Timothy, An extension of the Ap\'ery number supercongruence, Acta Arith., 123, 4, 335-348 (2006) · Zbl 1170.11008 [15] Koblitz, Neal, $$p$$-adic analysis: a short course on recent work, London Mathematical Society Lecture Note Series 46, 163 pp. (1980), Cambridge University Press, Cambridge-New York · Zbl 0439.12011 [16] Koblitz, Neal, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics 97, x+248 pp. (1993), Springer-Verlag, New York · Zbl 0804.11039 [17] Lennon, Catherine, Arithmetic and analytic properties of finite field hypergeometric functions, (no paging) pp. (2011), ProQuest LLC, Ann Arbor, MI [18] Lennon, Catherine, Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves, Proc. Amer. Math. Soc., 139, 6, 1931-1938 (2011) · Zbl 1281.11104 [19] Lennon, Catherine, Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold, J. Number Theory, 131, 12, 2320-2351 (2011) · Zbl 1277.11037 [20] Mortenson, Eric, Supercongruences for truncated $$_{n+1}\!F_n$$ hypergeometric series with applications to certain weight three newforms, Proc. Amer. Math. Soc., 133, 2, 321-330 (electronic) (2005) · Zbl 1152.11327 [21] McCarthy, Dermot, Extending Gaussian hypergeometric series to the $$p$$-adic setting, Int. J. Number Theory, 8, 7, 1581-1612 (2012) · Zbl 1253.33024 [22] McCarthy, Dermot, On a supercongruence conjecture of Rodriguez-Villegas, Proc. Amer. Math. Soc., 140, 7, 2241-2254 (2012) · Zbl 1354.11030 [23] McCarthy, Dermot, Transformations of well-poised hypergeometric functions over finite fields, Finite Fields Appl., 18, 6, 1133-1147 (2012) · Zbl 1276.11198 [24] McCarthy, Dermot, The trace of Frobenius of elliptic curves and the $$p$$-adic gamma function, Pacific J. Math., 261, 1, 219-236 (2013) · Zbl 1296.11079 [25] [McC7] Dermot McCarthy and Matthew Papanikolas, A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform, arXiv:1205.1006. · Zbl 1395.11078 [26] Rodriguez-Villegas, Fernando, Hypergeometric families of Calabi-Yau manifolds. Calabi-Yau varieties and mirror symmetry, Toronto, ON, 2001, Fields Inst. Commun. 38, 223-231 (2003), Amer. Math. Soc., Providence, RI · Zbl 1062.11038 [27] [Sage] W. A. Stein et al., Sage Mathematics Software (Version 4.8), The Sage Development Team, 2012, http://www.sagemath.org. [28] Vega, M. Valentina, Hypergeometric functions over finite fields and their relations to algebraic curves, Int. J. Number Theory, 7, 8, 2171-2195 (2011) · Zbl 1287.11138
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