Equidistribution of Solutions of Ternary Quadratic Congruences Modulo Prime Powers

2023/04/25 by Haldar, Anup
#11K36 #11L40 #11T23 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2304.12787

Abstract

Let p be a fixed odd prime and Q(x,y,z)=ax²+bxy+cy²+dxz+eyz+fz² be a fixed quadratic form in ℤ[x,y,z] which is non-degenerate in \mathbbF_p[x,y,z] and (a(4ac-b²),p)=1. Let (x₀,y₀,z₀) be a fixed point in ℤ³. We study the behavior of solutions (x,y,z) of congruences of the form Q(x,y,z)≡0\bmodq with q=pⁿ, where max\|x-x₀|,|y-y₀|,|z-z₀|\≤ 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→∞, under the condition N≥ q^(1)/(2)+ε.

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