\(_4F_3\)-Gaussian hypergeometric series and traces of Frobenius for elliptic curves. (English) Zbl 1506.11150
Summary: In this article, we obtain finite field analogues of classical summation identities connecting \(F_3\)-Appell series and \(_4F_3\)-classical hypergeometric series. As an application, we establish a new summation formula satisfied by the \(_4F_3\)-Gaussian hypergeometric series. We further express certain special values of \(_4F_3\)-Gaussian hypergeometric series in terms of traces of the Frobenius endomorphisms of certain families of elliptic curves. We also explicitly find some special values of \(_4F_3\)-Gaussian hypergeometric series.
MSC:
| 11T24 | Other character sums and Gauss sums |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
| 11F30 | Fourier coefficients of automorphic forms |
Keywords:
hypergeometric series over finite fields; Appell series; elliptic curves; Fourier coefficients of modular formsSoftware:
ecdataReferences:
| [1] | Ahlgren, S.: Gaussian hypergeometric series and combinatorial congruences, Symbolic computation, number theory, special functions, physics and combinatorics, Dev. Math. 4, Kluwer, Dodrecht (2001) · Zbl 1037.33016 |
| [2] | Ahlgren, S.; Ono, K., A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math., 518, 187-212 (2000) · Zbl 0940.33002 |
| [3] | Bailey, W., Generalized Hypergeometric Series (1935), Cambridge: Cambridge University Press, Cambridge · JFM 61.0406.01 |
| [4] | Bailey, W., On the sum of terminating \({_3}F_2(1)\), Quart. J. Math. Oxford Ser. (2), 4, 237-240 (1953) · Zbl 0051.30803 · doi:10.1093/qmath/4.1.237 |
| [5] | Barman, R.; Kalita, G., Certain values of Gaussian hypergeometric series and a family of algebraic curves, Int. J. Number Theory, 8, 4, 945-961 (2012) · Zbl 1266.11076 · doi:10.1142/S179304211250056X |
| [6] | Barman, R.; Kalita, G., Elliptic curves and special values of Gaussian hypergeometric series, J. Number Theory, 133, 9, 3099-3111 (2013) · Zbl 1295.11137 · doi:10.1016/j.jnt.2013.03.010 |
| [7] | Barman, R.; Kalita, G., Hypergeometric functions over \(F_q\) and traces of Frobenius for elliptic curves, Proc. Am. Math. Soc., 141, 10, 3403-3410 (2013) · Zbl 1319.11090 · doi:10.1090/S0002-9939-2013-11617-5 |
| [8] | Barman, R.; Tripathi, M., Certain transformations and special values of hypergeometric functions over finite fields, Ramanujan J., 57, 1277-1306 (2022) · Zbl 1490.33002 · doi:10.1007/s11139-021-00410-1 |
| [9] | Berndt, B.; Evans, R.; Williams, K., Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts (1998), New York: John Wiley & Sons, New York · Zbl 0906.11001 |
| [10] | Cremona, JE, Algorithms for Modular Elliptic Curves (1992), UK: Cambridge Univ Press, UK · Zbl 0758.14042 |
| [11] | Diamond, F.; Shurman, J., A First Course in Modular Forms (2005), New York: Springer, New York · Zbl 1062.11022 |
| [12] | Evans, R., Hypergeometric \({_3}F_2(1/4)\) evaluations over finite fields and Hecke eigenforms, Proc. Amer. Math. Soc., 138, 2, 517-531 (2010) · Zbl 1268.11158 · doi:10.1090/S0002-9939-09-10091-6 |
| [13] | Evans, R., Some mixed character sum identities of Katz, J. Number Theory, 179, 17-32 (2017) · Zbl 1418.11155 · doi:10.1016/j.jnt.2017.03.012 |
| [14] | Evans, R.; Greene, J., Clausen’s theorem and hypergeometric functions over finite fields, Finite Fields Appl., 15, 1, 97-109 (2009) · Zbl 1161.33005 · doi:10.1016/j.ffa.2008.09.001 |
| [15] | Evans, R.; Greene, J., Evaluations of hypergeometric functions over finite fields, Hiroshima Math. J., 39, 2, 217-235 (2009) · Zbl 1241.11138 · doi:10.32917/hmj/1249046338 |
| [16] | Evans, R.; Greene, J., A quadratic hypergeometric \({_2}F_1\) transformation over finite field, Proc. Am. Math. Soc., 145, 1071-1076 (2017) · Zbl 1403.11079 · doi:10.1090/proc/13303 |
| [17] | Evans, R.; Greene, J., Some mixed character sum identities of Katz II, Res. Number Theory, 3, 8, 14 (2017) · Zbl 1418.11156 |
| [18] | Frechette, S.; Ono, K.; Papanikolas, M., Gaussian hypergeometric functions and traces of Hecke operators, Int. Math. Res. Not., 60, 3233-3262 (2004) · Zbl 1088.11029 · doi:10.1155/S1073792804132522 |
| [19] | Frechette, S.; Swisher, H.; Tu, F-T, A cubic transformation formula for Appell-Lauricella hypergeometric functions over finite fields, Res. Number Theory, 4, 27, 27 (2018) · Zbl 1462.33011 · doi:10.1007/s40993-018-0119-9 |
| [20] | Fuselier, JG, Hypergeometric functions over \(F_p\) and relations to elliptic curves and modular forms, Proc. Am. Math. Soc., 138, 1, 109-123 (2010) · Zbl 1222.11058 · doi:10.1090/S0002-9939-09-10068-0 |
| [21] | Fuselier, J., Long, L., Ramakrishna, R., Swisher, H., Tu, F.: Hypergeometric functions over finite fields, Memoirs of the AMS, (2019) · Zbl 1443.11254 |
| [22] | Fuselier, JG; McCarthy, D., Hypergeometric type identities in the \(p\)-adic setting and modular forms, Proc. Am. Math. Soc., 144, 4, 1493-1508 (2016) · Zbl 1397.11078 · doi:10.1090/proc/12837 |
| [23] | Goodson, H., Hypergeometric functions and relations to Dwork hypersurfaces, Int. J. Number Theory, 13, 2, 439-485 (2017) · Zbl 1419.11135 · doi:10.1142/S1793042117500269 |
| [24] | Goodson, H., A complete hypergeometric point count formula for Dwork hypersurfaces, J. Number Theory, 179, 142-171 (2017) · Zbl 1418.11157 · doi:10.1016/j.jnt.2017.03.018 |
| [25] | Greene, J., Hypergeometric functions over finite fields, Trans. Am. Math. Soc., 301, 1, 77-101 (1987) · Zbl 0629.12017 · doi:10.1090/S0002-9947-1987-0879564-8 |
| [26] | Greene, J.; Stanton, D., A character sum evaluation and Gaussian hypergeometric series, J. Number Theory, 23, 1, 136-148 (1986) · Zbl 0588.10038 · doi:10.1016/0022-314X(86)90009-0 |
| [27] | He, B.: A finite field analogue for Appell series \(F_3\), (2017). arXiv:1704.03509v1 |
| [28] | He, B.; Li, L.; Zhang, R., An Appell series over finite fields, Finite Fields Appl., 48, 11, 289-305 (2017) · Zbl 1373.33023 · doi:10.1016/j.ffa.2017.08.007 |
| [29] | Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, Springer International Edition, Springer, (2005) · Zbl 0712.11001 |
| [30] | Kalita, G., Values of Gaussian hypergeometric series and their connections to algebraic curves, Int. J. Number Theory, 14, 1, 1-18 (2018) · Zbl 1428.11118 · doi:10.1142/S179304211850001X |
| [31] | Katz, N.: Exponential Sums and Differential Equations, Ann. of Math. Stud., vol. 124, Princeton Univ. Press, Princeton, NJ, (1990) · Zbl 0731.14008 |
| [32] | Kilbourn, T., An extension of the Apéry number supercongruence, Acta Arith., 123, 335-348 (2006) · Zbl 1170.11008 · doi:10.4064/aa123-4-3 |
| [33] | Knapp, A., Elliptic Curves (1992), USA: Princeton Univ Press, USA · Zbl 0804.14013 |
| [34] | Koike, M., Hypergeometric series over finite fields and Apéry numbers, Hiroshima Math. J., 22, 3, 461-467 (1992) · Zbl 0784.11057 · doi:10.32917/hmj/1206128497 |
| [35] | Lennon, C., 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 · doi:10.1016/j.jnt.2011.05.005 |
| [36] | Lennon, C., Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves, Proc. Am. Math. Soc., 139, 6, 1931-1938 (2011) · Zbl 1281.11104 · doi:10.1090/S0002-9939-2010-10609-3 |
| [37] | Li, L.; Li, X.; Mao, R., Appell series \(F_1\) over finite fields, Int. J. Number Theory, 14, 3, 727-738 (2018) · Zbl 1390.33026 · doi:10.1142/S179304211850046X |
| [38] | McCarthy, D., Transformations of well-poised hypergeometric functions over finite fields, Finite Fields Appl., 18, 6, 1133-1147 (2012) · Zbl 1276.11198 · doi:10.1016/j.ffa.2012.08.007 |
| [39] | McCarthy, D.; Osburn, R., A \(p\)-adic analogue of a formula of Ramanujan, Arch. Math. (Basel), 91, 492-504 (2008) · Zbl 1175.33004 · doi:10.1007/s00013-008-2828-0 |
| [40] | McCarthy, D.; Papanicolas, MA, A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform, Int. J. Number Theory, 11, 8, 2431-2450 (2015) · Zbl 1395.11078 · doi:10.1142/S1793042115501134 |
| [41] | Mortenson, E., A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, J. Number Theory, 99, 139-147 (2003) · Zbl 1074.11045 · doi:10.1016/S0022-314X(02)00052-5 |
| [42] | Mortenson, E., Supercongruences for truncated \({_{n+1}}F_n\)-hypergeometric series with applications to certain weight three newforms, Proc. Am. Math. Soc., 133, 2, 321-330 (2005) · Zbl 1152.11327 · doi:10.1090/S0002-9939-04-07697-X |
| [43] | Ono, K., Values of Gaussian hypergeometric series, Trans. Am. Math. Soc, 350, 1205-1223 (1998) · Zbl 0910.11054 · doi:10.1090/S0002-9947-98-01887-X |
| [44] | Sadek, M.; El-Sissi, N.; Shamsi, A.; Zamani, N., Evaluation of Gaussian hypergeometric series using Huff’s models of elliptic curves, Ramanujan J., 48, 2, 357-368 (2019) · Zbl 1473.11135 · doi:10.1007/s11139-018-0075-y |
| [45] | Salerno, A., Counting points over finite fields and hypergeometric functions, Funct. Approx. Comment. Math., 49, 1, 137-157 (2013) · Zbl 1295.11074 · doi:10.7169/facm/2013.49.1.9 |
| [46] | Silverman, J., The Arithmetic of Elliptic Curves (1982), Berlin: Springer- Verlag, Berlin |
| [47] | Tripathi, M.; Barman, R., A finite field analogoue of the Appell series \(F_4\), Res. Number Theory, 4, 35, 23 (2018) · Zbl 1428.33028 |
| [48] | Tripathi, M.; Barman, R., Certain product formulas and values of Gaussian hypergeometric series, Res. Number Theory, 6, 26, 29 (2020) · Zbl 1467.33004 |
| [49] | Tripathi, M.; Barman, R., Appell series over finite fields and Gaussian hypergeometric series, Res. Math. Sci., 8, 28 (2021) · Zbl 1468.33008 · doi:10.1007/s40687-021-00266-3 |
| [50] | Tripathi, M.; Saikia, N.; Barman, R., Appell’s hypergeometric series over finite fields, Int. J. Number Theory, 16, 4, 673-692 (2020) · Zbl 1440.33012 · doi:10.1142/S1793042120500347 |
| [51] | Vega, MV, Hypergeometric functions over finite fields and their relations to algebraic curves, Int. J. Number Theory, 7, 8, 2171-2195 (2011) · Zbl 1287.11138 · doi:10.1142/S1793042111004976 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
