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 $\hat{I}{»}_1{x}_1^2+\cdots +\hat{I}{»}_n{x}_n^2\equiv \hat{I}{»}_{n+1}\mathrm{mod}{p}^m$, where $p$ is a fixed odd prime, $\hat{I}{»}_1,...,\hat{I}{»}_{n+1}$ are integer coefficients such that $(\hat{I}{»}_1\cdots \hat{I}{»}_n,p)=1$ and $m\to \infty$. If $n\ge 6$, $p\ge 5$ and the coefficients are fixed and satisfy $\hat{I}{»}_1,...,\hat{I}{»}_n>0$ and $(\hat{I}{»}_{n+1},p)=1$ (inhomogeneous case), we obtain an asymptotic formula which is valid for integral solutions $(x_1,...,x_n)$ in cubes of side length at least $p^{(1/2+\epsilon )m}$, centered at the origin. If $n\ge 4$ and $\hat{I}{»}_{n+1}=0$ (homogeneous case), we prove a result of the same strength for coefficients $\hat{I}{»}_i$ which are allowed to vary with $m$. These results extend previous results of the first- and the third-named authors and N. Bag.

20 pages