scholarly journals CODES ASSOCIATED WITH Sp(4,q) AND EVEN-POWER MOMENTS OF KLOOSTERMAN SUMS

2009 ◽  
Vol 79 (3) ◽  
pp. 427-435 ◽  
Author(s):  
JI HYUN KIM
Keyword(s):  
Explicit Expression ◽  
Recursive Formula ◽  
Kloosterman Sums ◽  
Gauss Sum ◽  
Two Dimensional ◽  
Power Moment ◽  
Pless Power Moment Identity ◽  
Power Moments

AbstractHere we derive a recursive formula for even-power moments of Kloosterman sums or equivalently for power moments of two-dimensional Kloosterman sums. This is done by using the Pless power-moment identity and an explicit expression of the Gauss sum for Sp(4,q).

scholarly journals Infinite families of recursive formulas generating power moments of Kloosterman sums: O −(2n,2r) case

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Dae Kim
Keyword(s):  
Exponential Sums ◽  
Linear Codes ◽  
Kloosterman Sums ◽  
Orthogonal Groups ◽  
Power Moment ◽  
Binary Linear Codes ◽  
Maximal Parabolic Subgroup ◽  
Double Cosets ◽  
Recursive Formulas ◽  
Power Moments

AbstractIn this paper, we construct eight infinite families of binary linear codes associated with double cosets with respect to a certain maximal parabolic subgroup of the special orthogonal group SO −(2n, 2r). And we obtain four infinite families of recursive formulas for the power moments of Kloosterman sums and four those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of “Gauss sums” for the orthogonal groups O −(2n, 2r).

ESTIMATES FOR THE ERROR TERM IN A UNIFORM ASYMPTOTIC EXPANSION OF THE JACOBI POLYNOMIALS

2003 ◽  
Vol 01 (02) ◽  
pp. 213-241 ◽  
Author(s):  
R. WONG ◽  
Y.-Q. ZHAO
Keyword(s):  
Asymptotic Expansion ◽  
Explicit Expression ◽  
Canonical Form ◽  
Error Term ◽  
Jacobi Polynomials ◽  
Recursive Formula ◽  
Contour Integral ◽  
Integration By Parts ◽  
Uniform Asymptotic Expansion ◽  
Repeated Use

There are now several ways to derive an asymptotic expansion for [Formula: see text], as n → ∞, which holds uniformly for [Formula: see text]. One of these starts with a contour integral, involves a transformation which takes this integral into a canonical form, and makes repeated use of an integration-by-parts technique. There are two advantages to this approach: (i) it provides a recursive formula for calculating the coefficients in the expansion, and (ii) it leads to an explicit expression for the error term. In this paper, we point out that the estimate for the error term given previously is not sufficient for the expansion to be regarded as genuinely uniform for θ near the origin, when one takes into account the behavior of the coefficients near θ = 0. Our purpose here is to use an alternative method to estimate the remainder. First, we show that the coefficients in the expansion are bounded for [Formula: see text]. Next, we give an estimate for the error term which is of the same order as the first neglected term.

Supercharacters and mixed moments of Kloosterman sums

2018 ◽  
Vol 14 (04) ◽  
pp. 1023-1032 ◽  
Author(s):  
Ángel Chávez ◽  
George Todd
Keyword(s):  
Elliptic Curve ◽  
Recent Work ◽  
Kloosterman Sums ◽  
Fourth Degree ◽  
Mixed Power ◽  
Power Moments ◽  
Supercharacter Theory ◽  
Trace Of Frobenius

Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on [Formula: see text]. We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve.

An explicit expression for the distribution of the number of sides of the typical Poisson-Voronoi cell

2003 ◽  
Vol 35 (4) ◽  
pp. 863-870 ◽  
Author(s):  
Pierre Calka
Keyword(s):  
Distribution Function ◽  
Explicit Expression ◽  
Voronoi Tessellation ◽  
Voronoi Cell ◽  
Two Dimensional ◽  
Typical Cell

In this paper, we give an explicit expression for the distribution of the number of sides (or equivalently vertices) of the typical cell of a two-dimensional Poisson-Voronoi tessellation. We use this formula to give a table of numerical values of the distribution function.

FUSION MATRICES AND C<1 (QUASI) LOCAL CONFORMAL THEORIES

1990 ◽  
Vol 05 (14) ◽  
pp. 2721-2735 ◽  
Author(s):  
P. FURLAN ◽  
A. CH. GANCHEV ◽  
V.B. PETKOVA
Keyword(s):  
Explicit Expression ◽  
Local Theory ◽  
Partition Functions ◽  
Two Dimensional ◽  
Modular Invariant ◽  
General Properties ◽  
Structure Constants ◽  
6J Symbols

General properties of the fusion matrices and their explicit expression given by the Uq( sl (2)) quantum 6j-symbols are exploited to analyze some two dimensional c<1 conformal theories. The primary fields structure constants of the local theory, corresponding to the (A10, E6) modular invariant, and of the Z2 quasilocal analogs of the (A, D) and (A10, E6) series, are computed. The list of the Γ0(2) submodular invariant partition functions on the torus is extended.

GAUSS AND KLOOSTERMAN SUMS OVER RESIDUE RINGS OF ALGEBRAIC INTEGERS

2011 ◽  
Vol 07 (01) ◽  
pp. 101-114
Author(s):  
S. GURAK
Keyword(s):  
Classical Result ◽  
Classical Case ◽  
Rational Numbers ◽  
Kloosterman Sums ◽  
Gauss Sum ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
Rings Of Algebraic Integers ◽  
Sum Formula ◽  
Recent Evaluation

Let K be a field of degree n over Q, the field of rational numbers, with ring of integers O. Fix an integer m > 1, say with [Formula: see text] as a product of distinct prime powers, and let χ be a numerical character modulo m of conductor f(χ). Set ζm = exp (2πi/m) and let M be any ideal of O satisfying Tr M ⊆ mZ and N(1 + M) ⊆ 1 + f(χ)Z, where Tr and N are the trace and norm maps for K/Q. Then the Gauss sum [Formula: see text] is well-defined. If in addition N(1 + M) ⊆ 1 + mZ, then the Kloosterman sums [Formula: see text] are well-defined for any numerical character η ( mod m). The computation of GM(χ) and RM(η, z) is shown to reduce to their determination for m = pr, a power of a prime p, where M is comprised solely of ideals of K lying above p. In this setting we first explicitly determine GM(χ) for m = pr (r > 1) generalizing Mauclaire's classical result for K = Q. Relying on the recent evaluation of Kloosterman sums for prime powers in p-adic fields, we then proceed to compute the Kloosterman sums RM(η, z) here for m = pr (r > 1) when o(η) | p -1. This determination generalizes Salie's result in the classical case K = Q with o(η) = 1 or 2.

scholarly journals Seventh power moments of Kloosterman sums

2010 ◽  
Vol 175 (1) ◽  
pp. 349-362 ◽  
Author(s):  
Ronald Evans
Keyword(s):  
Kloosterman Sums ◽  
Power Moments

scholarly journals Galois representations attached to moments of Kloosterman sums and conjectures of Evans

2014 ◽  
Vol 151 (1) ◽  
pp. 68-120 ◽  
Author(s):  
Zhiwei Yun ◽  
Christelle Vincent
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Kloosterman Sums ◽  
Symmetric Power ◽  
Reductive Groups ◽  
Local Systems ◽  
Symmetric Powers ◽  
Exterior Powers ◽  
Power Moments ◽  
Tensor Powers

AbstractKloosterman sums for a finite field $\mathbb{F}_{p}$ arise as Frobenius trace functions of certain local systems defined over $\mathbb{G}_{m,\mathbb{F}_{p}}$. The moments of Kloosterman sums calculate the Frobenius traces on the cohomology of tensor powers (or symmetric powers, exterior powers, etc.) of these local systems. We show that when $p$ ranges over all primes, the moments of the corresponding Kloosterman sums for $\mathbb{F}_{p}$ arise as Frobenius traces on a continuous $\ell$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ that comes from geometry. We also give bounds on the ramification of these Galois representations. All of this is done in the generality of Kloosterman sheaves attached to reductive groups introduced by Heinloth, Ngô and Yun [Ann. of Math. (2) 177 (2013), 241–310]. As an application, we give proofs of conjectures of Evans [Proc. Amer. Math. Soc. 138 (2010), 517–531; Israel J. Math. 175 (2010), 349–362] expressing the seventh and eighth symmetric power moments of the classical Kloosterman sum in terms of Fourier coefficients of explicit modular forms. The proof for the eighth symmetric power moment conjecture relies on the computation done in Appendix B by C. Vincent.

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