L-functions of twisted exponential sums over finite fields
- PDF / 318,798 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 114 Downloads / 275 Views
L-functions of twisted exponential sums over finite fields Wei Cao1 · Shaofang Hong2 Received: 19 December 2018 / Accepted: 21 November 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let Fq be the finite field of q elements and χ1 , . . . , χn the multiplicative characters of Fq . Given a Laurent polynomial f (X ) ∈ Fq [x1±1 , . . . , xn±1 ], the corresponding L-function is defined to be L ∗ (χ1 , . . . , χn , f ; T ) = exp
∞ h=1
Sh∗ (χ1 , . . . , χn , f )
Th , h
where Sh∗ (χ1 , . . . , χn , f ) is the twisted exponential sum defined in the extension of Fq of degree h. In this paper, we obtain the explicit formulae for L ∗ (χ1 , . . . , χn , f ; T ) for the Laurent polynomials with full column rank exponent matrix in terms of padic gamma functions, which generalizes the results of Wan, Hong and Cao. We also evaluate the slopes of the reciprocal zeros and reciprocal poles of L ∗ (χ1 , . . . , χn , f ; T ) and determine the p-adic Newton polygons of the polynomials associated to the Lfunction L ∗ (χ1 , . . . , χn , f ; T ). Keywords Twisted exponential sum · L-function · p-Adic gamma function · Newton polygon Mathematics Subject Classification Primary 11M38 · 11T24 · 14G10
Research of W. Cao was partially supported by National Science Foundation of China (Grant No. 11871291), and sponsored by the K.C. Wong Magna Fund of Ningbo University. Research of S.F. Hong was partially supported by National Science Foundation of China (Grant No. 11771304) and by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.
B
Shaofang Hong [email protected]; [email protected]; [email protected] Wei Cao [email protected]
1
School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R. China
2
Mathematical College, Sichuan University, Chengdu 610064, P.R. China
123
W. Cao, S. F. Hong
1 Introduction Let p be a prime number and q = pr with r ∈ Z+ . Denote by Fq the finite field of q elements and Fq∗ the multiplicative group of Fq . Let f (x1 , . . . , xn ) ∈ Fq [x1±1 , . . . , xn±1 ] be a Laurent polynomial in which the variables are allowed to have negative degrees. Suppose that f has the sparse representation as a sum of m nonzero terms: f (x1 , . . . , xn ) =
m
d
d
a j x1 1 j · · · xn n j , a j ∈ Fq∗ .
(1.1)
j=1
The exponent matrix of f is defined to be the matrix D := (V1 , . . . , Vm ) ∈ Mn×m (Z) where V j = (d1 j , . . . , dn j )T ∈ Zn . We write X = (x1 , . . . , xn ) and d
d
X Vj = x1 1 j · · · xn n j for short. Then (1.1) can be expressed in concise form as f (X ) = mj=1 a j X V j . In particular, if D is a nonsingular square matrix, i.e., n = m and det(D) = 0, then f is called diagonal (cf. [14,18]). The exponent matrix has been used by Cao et al. to study the problems concerning polynomials over finite fields, such as counting rational points and evaluating exponential sums (cf. [4,6,20]). In this paper, we will use it to study the twisted exponential sum of a given Laurent poly
Data Loading...
