Powers of Gauss sums in quadratic fields

Twisted Monomial Gauss Sums modulo Prime Powers

We show that twisted monomial Gauss sums modulo prime powers can be evaluated explicitly once the power is sufficiently large.

Bounds of Gauss Sums in Finite Fields

We consider Gauss sums of the form Gn(a) = ∑ x∈Fpm χ(x) with a nontrivial additive character χ 6= χ0 of a finite field Fpm of pm elements of characteristic p. The classical bound |Gn(a)| ≤ (n−1)pm/2 becomes trivial for n ≥ pm/2 + 1. We show that, combining some recent bounds of HeathBrown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on |Gn(a)| which is...

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...