Equidistribution of Solutions of Ternary Quadratic Congruences Modulo Prime Powers

Abstract

Let $p$ be a fixed odd prime and $Q(x,y,z)=ax^2+bxy+cy^2+dxz+eyz+fz^2$ be a fixed quadratic form in $\mathbb{Z}[x,y,z]$ which is non-degenerate in $\mathbb{F}_p[x,y,z]$ and $(a(4ac-b^2),p)=1.$ Let $(x_0,y_0,z_0)$ be a fixed point in $\mathbb{Z}^3$. We study the behavior of solutions $(x,y,z)$ of congruences of the form $Q(x,y,z)\equiv0\bmod{q}$ with $q=p^n,$ where max$\{|x-x_0|,|y-y_0|,|z-z_0|\}\leq N$ and $(z,p)=1.$ In fact, we consider a smooth version of this problem and establish an asymptotic formula (thus the existence of such solutions) when $n\rightarrow\infty$, under the condition $N\geq q^{\frac{1}{2}+\varepsilon}$.