Quadratic Form Gauss Sums

Abstract

Let p be a prime, n, r positive integers, S an integer coprime to p. We let Q_r denote an r-dimensional integral quadratic form. For convenience, set e(x) = e^{2 pi i x}, where x is any rational number, i is the imaginary unit. Denote the quadratic Gauss sum by G(S;p^n). The evaluation of this sum was completed by Gauss in the early 19th century. Many proofs of these results have subsequently been obtained through a variety of methods. We are interested in the so called quadratic form Gauss sum, given by G(Q_r;S;p^n) - \sum_{x_1, x_2, ..., x_r}^{p^n-1} e(S/p^n * Q_r). Under certain assumptions on Q_r, we show how we may express G(Q_r;S;p^n) as a product of quadratic Gauss sums.