Double Gauss Sums

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Gauss sums are fundamental objects in number theory and arithmetic geometry. A number of generalizations of Gauss sums appear in the literature. They have extensive applications in many areas of mathematics and computer science. I plan to study the two dimensional quadratic Gauss sums (double Gauss sums) and their applications. My main objective is to evaluate the double Gauss sums explicitly. I aim to use their evaluations to determine a general formula for the number of solutions of certain quadratic equations over residue rings modulo prime powers. As a long term objective, upon completing the evaluations of double Gauss sums, I plan to study multi-dimensional quadratic Gauss sums, and apply them to find the number of solutions of corresponding quadratic equations over residue rings. I also plan to use my research results on double Gauss sums together with Siegel's mass formula (the local density approach) to find the number of representations of positive integers by quadratic forms with integer coefficients.