Quadratic Gauss sums over $\mathbb{Z}^n/c\mathbb{Z}^n$: explicit formulas, a duality theorem, and applications to Weil representations and cubic hypersurfaces over $\mathbb{F}_p$

Abstract

We provide explicit formulas for quadratic Gauss sums over $\mathbb{Z}^n/c\mathbb{Z}^n$, which generalize some of the existing formulas, e.g., Skoruppa and Zagier's (for $n=2$), and Iwaniec and Kowalski's (for arbitrary $n$). We then give four main applications. As the first application, we prove a duality theorem, which relates a sum over a subgroup of $\mathbb{Z}^n/c\mathbb{Z}^n$ to another sum over the orthogonal complement. This allows us to give explicit formulas for quadratic Gauss partial sums over certain subgroups. As the second application, we give an explicit formula for the coefficients of Weil representations of $\mathrm{Mp}_2(\mathbb{Z})$, which has the advantage, compared to Scheithauer's, Strömberg's, and Boylan and Skoruppa's formulas, that it involves neither local data nor limits of theta series. As the third application, we provide an explicit formula for the number of solutions of an arbitrary quadratic congruence in $n$ variables modulo a prime, as well as an efficient formula modulo a general integer. As the final application, we give a uniform formula for the number of solutions of the Diophantine equation $dxyz=Q(x,y,z)$ over $\mathbb{F}_p$, where $Q$ is an arbitrary integral quadratic form with some weak restrictions. This generalizes a result of Baragar regarding the Markoff equation. There is also a formula relating Hecke Gauss sums to sums over $\mathbb{Z}^n/c\mathbb{Z}^n$, an explicit formula for Hecke Gauss sums over quadratic fields, and an example of computing a Hecke Gauss sum over a cyclotomic field.