Quadratic Gauss Sums

A. Quadratic Reciprocity Via Gauss Sums

Note on the Quadratic Gauss Sums

Let p be an odd prime and {χ(m) = (m/p)}, m = 0,1, . . . ,p − 1 be a finite arithmetic sequence with elements the values of a Dirichlet character χ modp which are defined in terms of the Legendre symbol (m/p), (m,p)= 1. We study the relation between the Gauss and the quadratic Gauss sums. It is shown that the quadratic Gauss sumsG(k;p) are equal to the Gauss sums G(k,χ) that correspond to this ...

Gauss Sums & Representation by Ternary Quadratic Forms

This paper specifies some conditions as to when an integer m is locally represented by a positive definite diagonal integer-matrix ternary quadratic form Q at a prime p. We use quadratic Gauss sums and a version of Hensel’s Lemma to count how many solutions there are to the equivalence Q(~x) ≡ m (mod p) for any k ≥ 0. Given that m is coprime to the determinant of the Hessian matrix of Q, we can...

On Gauss-Jacobi sums

In this paper, we introduce a kind of character sum which simultaneously generalizes the classical Gauss and Jacobi sums, and show that this “Gauss-Jacobi sum” also specializes to the Kloosterman sum in a particular case. Using the connection to the Kloosterman sums, we obtain in some special cases the upper bound (the “Weil bound”) of the absolute values of the Gauss-Jacobi sums. We also discu...

Half Gauss Sums