Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Gauss sums and Kloosterman sums over residue rings of algebraic integers

Author: Ronald Evans
Journal: Trans. Amer. Math. Soc. 353 (2001), 4429-4445
MSC (2000): Primary 11L05, 11T24
Published electronically: June 27, 2001
MathSciNet review: 1851177
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


Let $\mathcal{O}$ denote the ring of integers of an algebraic number field of degree $m$ which is totally and tamely ramified at the prime $p$. Write $\zeta_q= \exp(2\pi i/q)$, where $q=p^r$. We evaluate the twisted Kloosterman sum

\begin{displaymath}\sum\limits_{\alpha\in(\mathcal{O}/q \mathcal{O})^*} \chi(N(\alpha)) \zeta_q^{T(\alpha)+z/N(\alpha)},\end{displaymath}

where $T$ and $N$ denote trace and norm, and where $\chi$ is a Dirichlet character (mod $q$). This extends results of Salié for $m=1$ and of Yangbo Ye for prime $m$ dividing $p-1.$ Our method is based upon our evaluation of the Gauss sum

\begin{displaymath}\sum\limits_{\alpha\in (\mathcal{O}/q\mathcal{O})^*} \chi(N(\alpha)) \zeta_q^{T(\alpha)},\end{displaymath}

which extends results of Mauclaire for $m=1$.

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

  • 1. J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Math. Studies, No. 120, Princeton University Press, Princeton, 1989. MR 90m:22041
  • 2. B. C. Berndt, R. J. Evans, and K. S. Williams, Gauss and Jacobi Sums, Wiley-Interscience, N.Y., 1998. MR 99d:11092
  • 3. H. Davenport and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. Reine Angew. Math. 172(1934), 151-182.
  • 4. W. Duke, On multiple Salié sums, Proc. Amer. Math. Soc. 114(1992), 623-625. MR 92f:11113
  • 5. R. J. Evans, Twisted hyper-Kloosterman sums over finite rings of integers, Proceedings of the Millennial Conference on Number Theory, University of Illinois (May 21-26, 2000), A K Peters, Natick, MA, to appear in 2001.
  • 6. T. Funakura, A generalization of the Chowla-Mordell theorem on Gaussian sums, Bull. London Math. Soc. 24(1992), 424-430. MR 93e:11094
  • 7. N. Katz, Gauss Sums, Kloosterman Sums, and Monodromy Groups, Annals of Math. Studies, No. 116, Princeton University Press, Princeton, 1988. MR 91a:11028
  • 8. N. Koblitz,$p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions, Springer-Verlag, N. Y., 1977. MR 57:5964
  • 9. J.-L. Mauclaire, Sommes de Gauss modulo $p^{\alpha }$ I, Proc. Jap. Acad. Ser. A 59(1983), 109-112. MR 85f:11062a
  • 10. J.-L. Mauclaire, Sommes de Gauss modulo $p^{\alpha }$ II, Proc. Jap. Acad. Ser. A 59(1983), 161-163. MR 85f:11062a
  • 11. W. Narkiewicz, Elementary and Analytic Theory of Numbers, Springer-Verlag, Berlin and PWN-Polish Scientific Publishers, Warsaw, 1990. MR 91h:11107
  • 12. R. Odoni, On Gauss sums $(\mathrm{mod} p^{n}), n \ge 2$, Bull. London Math. Soc. 5(1973), 325-327. MR 48:6020
  • 13. H. Salié, Über die Kloostermanschen Summen $S(u, v; q)$, Math. Z. 34(1932), 91-109.
  • 14. R. Smith, On $n$-dimensional Kloosterman sums, J. Number Theory 11 (1979), 324-343. MR 80i:11052
  • 15. L. C. Washington, Introduction to Cyclotomic Fields, 2nd edition, Springer-Verlag, N. Y., 1997. MR 97h:11130
  • 16. Y. Ye, A hyper-Kloosterman sum identity, Science in China (Series A) 41(1998), 1158-1162. MR 99m:11094
  • 17. Y. Ye, The lifting of an exponential sum to a cyclic algebraic number field of prime degree, Trans. Amer. Math. Soc. 350(1998), 5003-5015. MR 99b:11092
  • 18. Y. Ye, Personal communication, November, 1999.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11L05, 11T24

Retrieve articles in all journals with MSC (2000): 11L05, 11T24

Additional Information

Ronald Evans
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112

Received by editor(s): November 17, 1999
Received by editor(s) in revised form: January 4, 2001
Published electronically: June 27, 2001
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society