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Recently Garashuk and Lisonek evaluated Kloosterman sums K (a) modulo 4 over a finite field F3m in the case of even K (a). They posed it as an open problem to characterize elements a in F3m for which K (a) ≡ 1 (mod4) and K (a) ≡ 3 (mod4). In this paper, we will give an answer to this problem. The result allows us to count the number of elements a in F3m belonging to each of these two classes.

The divisor function over arithmetic progressions

Etienne Fouvry, Henryk Iwaniec, Nicholas Katz (1992)

Acta Arithmetica

The eleventh power character of 2.

P.A. Leonard, B.C. Mortimer (1976)

Journal für die reine und angewandte Mathematik

The exponent of class groups on congruence function fields

Manchar Madan, Daniel Madden (1977)

Acta Arithmetica

The factorization of $Q (L (x_{1}), . . ., L (x_{k}))$ over a finite field where $Q (x_{1}, . . ., x_{k})$ is of first degree and L(x) is linear

L. Carlitz, A. Long (1977)

Acta Arithmetica

The factorization of the derivative of Dickson polynomials.

María T. Acosta de Orozco, Javier Gómez-Calderón (1991)

Extracta Mathematicae

The Farey Series of polynomials over a finite field.

William A. Webb (1986)

Elemente der Mathematik

The fluctuations in the number of points on a family of curves over a finite field

Maosheng Xiong (2010)

Journal de Théorie des Nombres de Bordeaux

Let $l \geq 2$ be a positive integer, $𝔽_{q}$ a finite field of cardinality $q$ with $q \equiv 1 (mod l)$ . In this paper, inspired by [6, 3, 4] and using a slightly different method, we study the fluctuations in the number of $𝔽_{q}$ -points on the curve $ℂ_{F}$ given by the affine model $ℂ_{F} : Y^{l} = F (X)$ , where $F$ is drawn at random uniformly from the set of all monic $l$ -th power-free polynomials $F \in 𝔽_{q} [X]$ of degree $d$ as $d \to \infty$ . The method also enables us to study the fluctuations in the number of $𝔽_{q}$ -points on the same family of curves arising from the set of monic irreducible...

The formulas for the coefficients of the sum and product of p-adic integers with applications to Witt vectors

Kejian Xu, Zhaopeng Dai, Zongduo Dai (2011)

Acta Arithmetica

The fraction of subspaces of $GF {(q)}^{n}$ with a specified number of minimal weight vectors is asymptotically Poisson.

Bender, Edward A., Canfield, E.Rodney (1997)

The Electronic Journal of Combinatorics [electronic only]

The fundamental theorem of prehomogeneous vector spaces modulo $p^{m}$ (With an appendix by F. Sato)

Raf Cluckers, Adriaan Herremans (2007)

Bulletin de la Société Mathématique de France

For a number field $K$ with ring of integers $𝒪_{K}$ , we prove an analogue over finite rings of the form $𝒪_{K} / 𝒫^{m}$ of the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where $𝒫$ is a big enough prime ideal of $𝒪_{K}$ and $m > 1$ . In the appendix, F.Sato gives an application of the Theorems 1.1, 1.3 and the Theorems A, B, C in J.Denef and A.Gyoja [Character sums associated to prehomogeneous vector spaces, Compos. Math., 113(1998), 237–346] to the functional equation of $L$ -functions...

The $G L (n, F_{p})$ -invariance of the Potts Hamiltonian.

Caragiu, Mihai, Caragiu, Mellita (1997)

International Journal of Mathematics and Mathematical Sciences

The geometry of the third moment of exponential sums

Florent Jouve (2008)

Journal de Théorie des Nombres de Bordeaux

We give a geometric interpretation (and we deduce an explicit formula) for two types of exponential sums, one of which is the third moment of Kloosterman sums over $F_{q}$ of type $K (ν^{2}; q)$ . We establish a connection between the sums considered and the number of $F_{q}$ -rational points on explicit smooth projective surfaces, one of which is a $K 3$ surface, whereas the other is a smooth cubic surface. As a consequence, we obtain, applying Grothendieck-Lefschetz theory, a generalized formula for the third moment of Kloosterman...

The irreducibility of compositions of linear polynomials over a finite field

S. D. Cohen (1982)

Compositio Mathematica

The joint distribution of $Q$ -additive functions on polynomials over finite fields

Michael Drmota, Georg Gutenbrunner (2005)

Journal de Théorie des Nombres de Bordeaux

Let $K$ be a finite field and $Q \in K [T]$ a polynomial of positive degree. A function $f$ on $K [T]$ is called (completely) $Q$ -additive if $f (A + B Q) = f (A) + f (B)$ , where $A, B \in K [T]$ and $deg (A) < deg (Q)$ . We prove that the values $(f_{1} (A), ..., f_{d} (A))$ are asymptotically equidistributed on the (finite) image set ${(f_{1} (A), ...,$ $f_{d} ...$

The number of matrix fields over GF(q)

J. Beard (1974)

Acta Arithmetica

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