Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

   
 
 

 

A conjecture of Evans on sums of Kloosterman sums


Authors: Evan P. Dummit, Adam W. Goldberg and Alexander R. Perry
Journal: Proc. Amer. Math. Soc. 138 (2010), 3047-3056
MSC (2010): Primary 11L05; Secondary 33C20
DOI: https://doi.org/10.1090/S0002-9939-10-10486-9
Published electronically: May 4, 2010
MathSciNet review: 2653929
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. S. Ahlgren, Gaussian hypergeometric series and combinatorial congruences, Symbolic computation, number theory, special functions, physics and combinatorics [Eds. F. Garvan and M. E. H. Ismail], Kluwer Acad. Publ. Dordrecht (2001), 1-12. MR 1880076 (2003i:33025)
  • 2. S. Ahlgren and K. Ono, Modularity of a certain Calabi-Yau threefold, Monatsh. Math. 129 (2000), 177-190. MR 1746757 (2001b:11059)
  • 3. S. Ahlgren and K. Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math 518 (2000), 187-212. MR 1739404 (2001c:11057)
  • 4. R. Apéry, Irrationalité de $ \zeta(2)$ et $ \zeta(3)$, Astérisque 61 (1979), 11-13.
  • 5. R. J. Evans, Hypergeometric $ {}_3F_2(1/4)$ evaluations over finite fields and Hecke eigenforms, Proc. Amer. Math. 138 (2010), 517-531. MR 2557169
  • 6. R. J. Evans and J. Greene, Clausen's theorem and hypergeometric functions over finite fields, Finite Fields Appl. 15 (2009), 97-109. MR 2468995 (2010a:33061)
  • 7. J. Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), 77-101. MR 879564 (88e:11122)
  • 8. K. Ireland and M. Rosen, A classical introduction to modern number theory, Springer-Verlag, New York, 1990. MR 1070716 (92e:11001)
  • 9. D. H. Lehmer and Emma Lehmer, On the cubes of Kloosterman sums, Acta Arith. 6 (1960), 15-22. MR 0115976 (22:6773)
  • 10. L. J. Mordell, On Lehmer's congruence associated with cubes of Kloosterman's sums, Journal London Math. Soc. 36 (1961), 335-339. MR 0126421 (23:A3717)
  • 11. K. Ono, The web of modularity: Arithmetic of the coefficients of modular forms and q-series, CBMS, vol. 102, Amer. Math. Soc., Providence, RI, 2004. MR 2020489 (2005c:11053)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11L05, 33C20

Retrieve articles in all journals with MSC (2010): 11L05, 33C20


Additional Information

Evan P. Dummit
Affiliation: Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, Wisconsin 53706-1388
Email: Dummit@math.wisc.edu

Adam W. Goldberg
Affiliation: 617 Logan Lane, Danville, California 94526
Email: AdamWGoldberg@gmail.com

Alexander R. Perry
Affiliation: Department of Mathematics, 4517 Lerner Hall, Columbia University, 2920 Broadway, New York, New York 10027-8343
Email: arp2125@columbia.edu

DOI: https://doi.org/10.1090/S0002-9939-10-10486-9
Received by editor(s): July 24, 2009
Received by editor(s) in revised form: July 27, 2009
Published electronically: May 4, 2010
Communicated by: Jim Haglund
Article copyright: © Copyright 2010 Evan Dummit, Adam Goldberg, Alexander Perry

American Mathematical Society