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ISSN 1088-6850(e) ISSN 0002-9947(p)

Gauss sums and Kloosterman sums over residue rings of algebraic integers

Author(s): Ronald Evans
Journal: Trans. Amer. Math. Soc. 353 (2001), 4429-4445.
MSC (2000): Primary 11L05, 11T24
Posted: June 27, 2001
MathSciNet review: 1851177
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Abstract | References | Similar articles | Additional information

Abstract:

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

and

denote trace and norm, and where $\chi$ is a Dirichlet character (mod

). This extends results of Salié for

and of Yangbo Ye for prime

dividing

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

References:

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