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$ L$-functions of twisted diagonal exponential sums over finite fields

Author: Shaofang Hong
Journal: Proc. Amer. Math. Soc. 135 (2007), 3099-3108
MSC (2000): Primary 11L03, 11T23, 14G10
Published electronically: June 20, 2007
MathSciNet review: 2322739
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Abstract: Let $ {\bf F}_q$ be the finite field of $ q$ elements with characteristic $ p$ and $ {\bf F}_{q^m}$ its extension of degree $ m$. Fix a nontrivial additive character $ \Psi$ and let $ \chi _1,..., \chi _n$ be multiplicative characters of $ {\bf F}_p.$ For

$\displaystyle f(x_1,...,x_n) \in {\bf F}_q[x_1,x_1^{-1},...,x_n,x^{-1}_n],$

one can form the twisted exponential sum $ S^*_m(\chi _1,...,\chi _n,f)$. The corresponding $ L$-function is defined by

$\displaystyle L^*(\chi _1,..., \chi _n,f;t)=\hbox {exp}(\sum^{\infty}_{m=0}S^*_m(\chi _1,...,\chi _n, f){\frac {t^m} {m}} ).$

In this paper, by using the $ p$-adic gamma function and the Gross-Koblitz formula on Gauss sums, we give an explicit formula for the $ L$-function $ L^*(\chi _1,...,\chi _n, f;t)$ if $ f$ is a Laurent diagonal polynomial. We also determine its $ p$-adic Newton polygon.

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Additional Information

Shaofang Hong
Affiliation: Mathematical College, Sichuan University, Chengdu 610064, People’s Republic of China

Keywords: Twisted exponential sum, $L$-function, $p$-adic Newton polygon, $\Gamma$-function, Gross--Koblitz formula, Hasse--Davenport relation.
Received by editor(s): May 1, 2006
Received by editor(s) in revised form: July 20, 2006
Published electronically: June 20, 2007
Additional Notes: The research of this author was supported by New Century Excellent Talents in University Grant # NCET-060785, by SRF for ROCS, SEM and by NNSF of China Grant # 10101015
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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