Asymptotic behavior of small solutions of quadratic congruences in three variables modulo prime powers

Abstract

Let $p>5$ be a fixed prime and assume that $\hat{I}{\pm }_1,\hat{I}{\pm }_2,\hat{I}{\pm }_3$ are coprime to $p$. We study the asymptotic behavior of small solutions of congruences of the form $\hat{I}{\pm }_1{x}_1^2+\hat{I}{\pm }_2{x}_2^2+\hat{I}{\pm }_3{x}_3^2\equiv 0\mathrm{mod}q$ with $q=p^n$, where $\max \{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_1{x}_2{x}_3,p)=1$. (In fact, we consider a smoothed version of this problem.) If $\hat{I}{\pm }_1,\hat{I}{\pm }_2,\hat{I}{\pm }_3$ are fixed and $n\to \infty$, we establish an asymptotic formula (and thereby the existence of such solutions) under the condition $N\gg q^{1/2+\epsilon }$. If these coefficients are allowed to vary with $n$, we show that this formula holds if $N\gg q^{11/18+\epsilon }$. The latter should be compared with a result by Heath-Brown who established the existence of non-zero solutions under the condition $N\gg q^{5/8+\epsilon }$ for odd square-free moduli $q$.

20 pages. arXiv admin note: substantial text overlap with arXiv:2201.05871