Multiple exponential sums and their applications to quadratic congruences

TL;DR

The paper develops a multivariable theory for exponential sums with rational amplitudes $f=f_1/f_2$ over $n$ variables modulo $p^m$, extending single-variable results to higher dimensions. It derives explicit, parity-dependent evaluations of $S_{\boldsymbol{\alpha}}(f,p^m)$ under gradient/Hessian conditions and provides lifting results for critical points, enabling precise control of exponential sums in multiple variables. The main application counts weighted solutions to quadratic congruences $Q(x_1,\dots,x_n)\equiv 0\pmod{p^m}$ inside small boxes via a multivariable Poisson summation, yielding an asymptotic $T\sim b_p\cdot N^n/p^m$ for large box sizes with a positive density $b_p$; this extends prior one-variable results and establishes equidistribution of such solutions in the multivariate setting.

Abstract

In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for $n\geq 2$, a fixed natural number, we obtain an asymptotic formula for the (weighted) number of solutions of quadratic congruences of the form $x_1^2+x_2^2+...+x_n^2\equiv x_{n+1}^2\bmod{p^m}$ in small boxes, thus establishing an equidistribution result for these solutions.

Theorems & Definitions (8)

• Proposition 1
• Theorem 1
• Theorem 2
• Remark 1
• Proposition 2: Poisson summation formula
• proof
• Proposition 3: Heath-Brown
• proof