Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers

TL;DR

This paper establishes asymptotic formulas for the number of small solutions to quadratic congruences modulo prime powers, extending previous results to broader cases with fixed or varying coefficients.

Contribution

It provides new asymptotic formulas for solutions of quadratic congruences modulo prime powers, including inhomogeneous and homogeneous cases with variable coefficients.

Findings

Asymptotic formulas for solutions in the inhomogeneous case with fixed coefficients.

Asymptotic formulas for solutions in the homogeneous case with variable coefficients.

Results valid for solutions within cubes of side length at least p^{(1/2+ε)m}.

Abstract

We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form $\lambda_1x_1^2+\cdots +\lambda_nx_n^2\equiv \lambda_{n+1}\bmod{p^m}$, where $p$ is a fixed odd prime, $\lambda_1,...,\lambda_{n+1}$ are integer coefficients such that $(\lambda_1\cdots \lambda _{n},p)=1$ and $m\rightarrow \infty$. If $n\ge 6$, $p\ge 5$ and the coefficients are fixed and satisfy $\lambda_1,...,\lambda_n>0$ and $(\lambda_{n+1},p)=1$ (inhomogeneous case), we obtain an asymptotic formula which is valid for integral solutions $(x_1,...,x_n)$ in cubes of side length at least $p^{(1/2+\varepsilon)m}$, centered at the origin. If $n\ge 4$ and $\lambda_{n+1}=0$ (homogeneous case), we prove a result of the same strength for coefficients $\lambda_i$ which are allowed to vary with $m$. These results extend previous results of the first- and the third-named authors and N. Bag.