Abstract
For primes \(p>3\) we give a formula for the traces of pth Hecke operators on space of cusp forms of weight k and level 1 involving p-adic hypergeometric functions. We also express the traces of Frobenius for a certain family of elliptic curves in terms of p-adic hypergeometric function. Finally, we derive certain relations for Ramanujan’s \(\tau \)-function and p-adic hypergeometric functions.
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References
Ahlgren, S.: The points of a certain fivefold over finite fields and the twelfth power of the eta function. Finite Fields Appl. 8(1), 18–33 (2002)
Ahlgren, S., Ono, K.: Modularity of a certain Calabi–Yau threefold. Monatshefte Math. 129(3), 177–190 (2000)
Barman, R., Kalita, G.: hypergeometric functions over \({\mathbb{F}}_q\) and traces of Frobenius for elliptic curves. Proc. Am. Math. Soc. 141(10), 3403–3410 (2013)
Barman, R., Saikia, N., McCarthy, D.: Summation identities and special values of hypergeometric series in the \(p\)-adic setting. J. Number Theory 153, 63–84 (2015)
Berndt, B., Evans, R., Williams, K.: Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication. Wiley, New York (1998)
Frechette, S., Ono, K., Papanikolas, M.: Gaussian hypergeometric functions and traces of Hecke operators. Int. Math. Res. Not. 60 (2004)
Frechette, S., Ono, K., Papanikolas, M.: The combinatorics of traces of Hecke operators. Proc. Natl. Acad. Sci. 101, 17016–17020 (2004)
Fuselier, J.: Hypergeometric functions over finite fields and relations to modular forms and elliptic curves, Ph.D. thesis, Texas A&M University (2007)
Fuselier, J.: Hypergeometric functions over \({\mathbb{F}}_p\) and relations to elliptic curves and modular forms. Proc. Am. Math. Soc. 138(1), 109–123 (2010)
Fuselier, J.: Traces of Hecke operators in level 1 and Gaussian hypergeometric functions. Proc. Am. Math. Soc. 141(6), 1871–1881 (2013)
Greene, J.: Hypergeometric functions over finite fields. Trans. Am. Math. Soc. 301(1), 77–101 (1987)
Gross, B.H., Koblitz, N.: Gauss sum and the \(p\)-adic \(\Gamma \)-function. Ann. Math. 109, 569–581 (1979)
Hijikata, H., Pizer, A., Shemanske, T.: The basic problems for modular forms on \(\Gamma _0(N)\). Memoirs Am. Math. Soc. 82 (1989)
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Springer International Edition. Springer, New York (2005)
Koblitz, N.: \(p\)-Adic Analysis: A Short Course on Recent Work. London Mathematical Society Lecture Note Series, vol. 46. Cambridge University Press, Cambridge (1980)
Lennon, C.: Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves. Proc. Am. Math. Soc. 139, 931–938 (2011)
McCarthy, D.: Extending Gaussian hypergeometric series to the \(p\)-adic setting. Int. J. Number Theory 8(7), 1581–1612 (2012)
McCarthy, D.: The trace of Frobenius of elliptic curves and the \(p\)-adic gamma function. Pac. J. Math. 261(1), 219–236 (2013)
Ono, K.: Values of Gaussian hypergeometric series. Trans. Am. Math. Soc. 350(3), 1205–1223 (1998)
Rouse, J.: Vanishing and non-vanishing of traces of Hecke operators. Trans. Am. Math. Soc. 358(10), 4637–4651 (2006)
Schoof, R.: Nonsingular plane cubic curves over finite fields. J. Comb. Theory Ser. A 46, 183–211 (1987)
Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. (N.S.) 20, 47–87 (1956)
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Funding was provided by the Department of Science and Technology, Ministry of Science and Technology (Grant No. IFA16-MA77). Neelam Saikia acknowledges the financial support of the Department of Science and Technology, Government of India for supporting a part of this work under INSPIRE Faculty Fellowship.
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Pujahari, S., Saikia, N. Traces of Hecke operators in level 1 and p-adic hypergeometric functions. Ramanujan J 52, 519–539 (2020). https://doi.org/10.1007/s11139-019-00170-z
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DOI: https://doi.org/10.1007/s11139-019-00170-z
Keywords
- Traces of Hecke operators
- Elliptic curves
- Traces of Frobenius
- p-adic hypergeometric functions
- Gauss sums
- p-adic Gamma function