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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 $\mathbb{F}_p$

Xiao-Jie Zhu

TL;DR

The work develops explicit, multi-variable quadratic Gauss-sum formulas over $\mathbb{Z}^n/c\mathbb{Z}^n$ and uses finite-quadratic-module and Fourier-analytic techniques to derive a main theorem, a duality for subsums, and explicit Weil-representation coefficients without local data. These results yield concrete applications: a duality theorem for Gauss subsums, explicit coefficients of Weil representations, and uniform counts for affine quadratic hypersurfaces and generalized Markoff-type equations over finite fields. The methods connect lattice theory, Milgram-type extensions, and Hecke Gauss sums, with broad implications for vector-valued modular forms, theta series, and Diophantine counting problems. Collectively, the paper provides a toolkit for explicit arithmetic in quadratic forms and their representations, with wide-reaching theoretical and computational consequences.

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.

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 $\mathbb{F}_p$

TL;DR

The work develops explicit, multi-variable quadratic Gauss-sum formulas over and uses finite-quadratic-module and Fourier-analytic techniques to derive a main theorem, a duality for subsums, and explicit Weil-representation coefficients without local data. These results yield concrete applications: a duality theorem for Gauss subsums, explicit coefficients of Weil representations, and uniform counts for affine quadratic hypersurfaces and generalized Markoff-type equations over finite fields. The methods connect lattice theory, Milgram-type extensions, and Hecke Gauss sums, with broad implications for vector-valued modular forms, theta series, and Diophantine counting problems. Collectively, the paper provides a toolkit for explicit arithmetic in quadratic forms and their representations, with wide-reaching theoretical and computational consequences.

Abstract

We provide explicit formulas for quadratic Gauss sums over , which generalize some of the existing formulas, e.g., Skoruppa and Zagier's (for ), and Iwaniec and Kowalski's (for arbitrary ). We then give four main applications. As the first application, we prove a duality theorem, which relates a sum over a subgroup of 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 , 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 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 over , where 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 , an explicit formula for Hecke Gauss sums over quadratic fields, and an example of computing a Hecke Gauss sum over a cyclotomic field.

Paper Structure

This paper contains 38 sections, 51 theorems, 277 equations.

Table of Contents

  1. Introduction
  2. The main theorem
  3. A duality theorem
  4. A duality theorem
  5. A formula for Weil representations
  6. Affine quadratic hypersurfaces over $\mathbb{F}_p$
  7. Generalized Markoff equations over $\mathbb{F}_p$
  8. Other results, and notations
  9. Weil representations and finite quadratic modules
  10. Lattices and Weil representations
  11. Finite quadratic modules
  12. Fourier analysis on finite abelian groups
  13. Two auxiliary Gauss sums and an extension of Milgram's formula
  14. Basic properties of $\mathfrak{G}_{G}(r,\mathbf{t};\mathbf{w},\mathbf{x})$
  15. Two auxiliary Gauss sums
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $G$ be a positive definite even integral symmetric matrix of size $n\in\mathbb{Z}_{\geq 1}$. Let $c$ be a positive integer and $a$ be a nonzero integer coprime to $c$. Let $D=\det(G)$ and $N$ be the level of $G$ (which means the least positive integer such that $N\cdot G^{-1}$ is even integral). Then: (1) If $\gcd(N,c)=1$, we have where $c_0$ is the odd part of $c$, and $c'$ is any solution o

Theorems & Definitions (116)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1: Shintani
  • Example 2.2
  • Lemma 2.3
  • proof
  • ...and 106 more