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Transactions of the American Mathematical Society
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
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Abstract:

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 $m=1$.


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Additional Information

Ronald Evans
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112
Email: revans@ucsd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02823-9
PII: S 0002-9947(01)02823-9
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