A conjecture of Evans on sums of Kloosterman sums

Dummit, Evan; Goldberg, Adam; Perry, Alexander

doi:10.1090/S0002-9939-10-10486-9

A conjecture of Evans on sums of Kloosterman sums
HTML articles powered by AMS MathViewer

by Evan P. Dummit, Adam W. Goldberg and Alexander R. Perry

Proc. Amer. Math. Soc. 138 (2010), 3047-3056

DOI: https://doi.org/10.1090/S0002-9939-10-10486-9

Published electronically: May 4, 2010

PDF

Abstract:

In a recent paper, Evans relates twisted Kloosterman sheaf sums to Gaussian hypergeometric functions, and he formulates a number of conjectures relating certain twisted Kloosterman sheaf sums to the coefficients of modular forms. Here we prove one of his conjectures for a fourth order twisted Kloosterman sheaf sum $T_n$ of the quadratic character on $\mathbf {F}_p^\times$. In the course of the proof we develop reductions for twisted moments of Kloosterman sums and apply these in the end to derive a congruence relation for $T_n$ with generalized Apéry numbers.

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, Modularity of a certain Calabi-Yau threefold, Monatsh. Math. 129 (2000), no. 3, 177–190. MR 1746757, DOI 10.1007/s006050050069
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
R. Apéry, Irrationalité de $\zeta (2)$ et $\zeta (3)$, Astérisque 61 (1979), 11–13.
Ron Evans, Hypergeometric $_3F_2(1/4)$ evaluations over finite fields and Hecke eigenforms, Proc. Amer. Math. Soc. 138 (2010), no. 2, 517–531. MR 2557169, DOI 10.1090/S0002-9939-09-10091-6
Ron Evans and John Greene, Clausen’s theorem and hypergeometric functions over finite fields, Finite Fields Appl. 15 (2009), no. 1, 97–109. MR 2468995, DOI 10.1016/j.ffa.2008.09.001
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
Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
D. H. Lehmer and Emma Lehmer, On the cubes of Kloosterman sums, Acta Arith. 6 (1960), 15–22. MR 115976, DOI 10.4064/aa-6-1-15-22
L. J. Mordell, On Lehmer’s congruence associated with cubes of Kloosterman’s sums, J. London Math. Soc. 36 (1961), 335–339. MR 126421, DOI 10.1112/jlms/s1-36.1.335
Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489