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 \(\alpha _1,\alpha _2,\alpha _3\) are coprime to p. We study the asymptotic behavior of small solutions of congruences of the form \(\alpha _1x_1^2+\alpha _2x_2^2+\alpha _3x_3^2\equiv 0\bmod {q}\) with \(q=p^n\), where \(\max \{|x_1|,|x_2|,|x_3|\}\le N\) and \((x_1x_2x_3,p)=1\). (In fact, we consider a smoothed version of this problem.) If \(\alpha _1,\alpha _2,\alpha _3\) are fixed and \(n\rightarrow \infty \), we establish an asymptotic formula (and thereby the existence of such solutions) under the condition \(N\gg q^{1/2+\varepsilon }\). If these coefficients are allowed to vary with n, we show that this formula holds if \(N\gg q^{11/18+\varepsilon }\). The latter should be compared with a result by Heath-Brown who established the existence of non-zero solutions under the condition \(N \gg q^{5/8+\varepsilon }\) for odd square-free moduli q.