Double Gauss Sums

高斯和是数论和算术几何中的基本对象。文献中出现了许多高斯和的推广。它们在数学和计算机科学的许多领域都有广泛的应用。我计划研究二维二次高斯和（双高斯和）及其应用。我的主要目标是明确地计算双高斯和。我的目的是利用他们的评价来确定残环上模素幂的某些二次方程解的个数的一般公式。作为一个长期目标，在完成双高斯和的评价后，我计划研究多维二次高斯和，并将其应用于求残环上对应的二次方程的解的个数。我还打算利用我的双高斯和的研究成果，结合西格尔的质量公式（局部密度法），找出正整数用整数系数的二次型表示的个数。

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.