Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Elementary estimates for a certain typeof Soto-Andrade sum


Author: Imin Chen
Journal: Proc. Amer. Math. Soc. 128 (2000), 1933-1939
MSC (2000): Primary 11L40; Secondary 05C25, 20G40
Published electronically: February 21, 2000
MathSciNet review: 1707143
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

This paper shows that a certain type of Soto-Andrade sum can be estimated in an elementary way which does not use the Riemann hypothesis for curves over finite fields and which slightly sharpens previous estimates for this type of Soto-Andrade sum. As an application, we discuss how this implies that certain graphs arising from finite upper half planes in odd characteristic are Ramanujan without using the Riemann hypothesis.


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

  • 1. Nancy Celniker, Steven Poulos, Audrey Terras, Cindy Trimble, and Elinor Velasquez, Is there life on finite upper half planes?, A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math., vol. 143, Amer. Math. Soc., Providence, RI, 1993, pp. 65–88. MR 1210513 (94h:05055), http://dx.doi.org/10.1090/conm/143/00991
  • 2. Nicholas M. Katz, Estimates for Soto-Andrade sums, J. Reine Angew. Math. 438 (1993), 143–161. MR 1215651 (94h:11109), http://dx.doi.org/10.1515/crll.1993.438.143
  • 3. Wen-Ch'ing Winnie Li.
    Number-theoretic constructions of ramanujan graphs.
    Astérisque, 3-4(228):101-120, 1995.
  • 4. Wen-Ching Winnie Li.
    Estimates of character sums arising from finite upper half planes.
    In Cohen S. and Niederreiter H., editors, Finite fields and applications (Glasgow, 1995), number 233 in London Mathematical Society Lecture Notes. Cambridge University Press, 1996.
  • 5. Yoshiaki Sawabe, Legendre character sums, Hiroshima Math. J. 22 (1992), no. 1, 15–22. MR 1160036 (93i:11147)
  • 6. J. Soto-Andrade.
    Geometrical Gelfand models, tensor quotients, and Weil representations.
    In Proceedings of Symposia in Pure Mathematics, volume 47, pages 305-316, 1987.
  • 7. A. Terras.
    Are finite upper half plane graphs Ramanujan?
    In Expanding Graphs (Princeton, NJ, 1992), volume 10 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 125-142. American Mathematical Society, Providence, RI, 1993. CMP 93:17
  • 8. André Weil, On some exponential sums, Proc. Nat. Acad. Sci. U. S. A. 34 (1948), 204–207. MR 0027006 (10,234e)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11L40, 05C25, 20G40

Retrieve articles in all journals with MSC (2000): 11L40, 05C25, 20G40


Additional Information

Imin Chen
Affiliation: Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 2K6
Email: chen@math.mcgill.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05591-X
PII: S 0002-9939(00)05591-X
Received by editor(s): September 8, 1998
Published electronically: February 21, 2000
Additional Notes: This research was supported by an NSERC postdoctoral fellowship and a grant from CICMA
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2000 American Mathematical Society