Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers

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.