Small Solutions of generic ternary quadratic congruences
Stephan Baier, Aishik Chattopadhyay
TL;DR
The paper studies small solutions to the diagonal ternary quadratic congruence $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0\bmod q$ with odd prime-power modulus $q=p^m$, fixed $(α_2,q)=1$, and varying $α_3$ coprime to $q$. Using a variance method that detects the congruence via Dirichlet characters, the authors decompose the count into a main term plus an error, and bound the error by combining Heath-Brown’s, Burgess’s, and Milicević’s sum estimates along with Gauss-sum techniques and p-adic exponent-pair optimization. They derive an explicit asymptotic for the solution count: $S(α_3)=C_q\frac{N^3}{q}igl(1+o(1)\bigr)$ for all but $o(φ(q))$ admissible $α_3$, with the main term constant $C_q=8\left(1-\frac{1}{p}\right)\left(1-\frac{1}{p}\left(\frac{-α_2}{p}\right)\right)$. The results yield several exponent improvements: unconditionally $N\ge q^{11/24+ε}$, a stronger $11/25$ bound when $q=p^m$ with fixed $p$ and $m\to\infty$, and a $1/3$-exponent under the Lindelöf hypothesis. This work breaks the $1/2$ barrier for generic diagonal ternary forms and clarifies the role of exceptional $α_3$ and $p$-adic phenomena in small-solution problems.
Abstract
We consider small solutions of quadratic congruences of the form $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$, where $q=p^m$ is an odd prime power. Here, $α_2$ is arbitrary but fixed and $α_3$ is variable, and we assume that $(α_2α_3,q)=1$. We show that for all $α_3$ modulo $q$ which are coprime to $q$ except for a small number of $α_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$ with $\max\{|x_1|,|x_2|,|x_3|\}\le N$ holds if $N\ge q^{11/24+\varepsilon}$ as $q$ tends to infinity over the set of all odd prime powers. It is of significance that we break the barrier 1/2 in the above exponent. If $q$ is restricted to powers $p^m$ of a {\it fixed} prime $p$ and $m$ tends to infinity, we obtain a slight improvement of this result using the theory of $p$-adic exponent pairs, as developed by Milićević, replacing the exponent $11/24$ above by $11/25$. Under the Lindelöf hypothesis for Dirichlet $L$-functions, we are able to replace the exponent $11/24$ above by $1/3$.