Title: Small Solutions of generic ternary quadratic congruences Show Chinese title

Title: 关于一般三元二次同余式的微小解

Abstract: We consider small solutions of quadratic congruences of the form $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$, where $q=p^m$ is an odd prime power. Here, $\alpha_2$ is arbitrary but fixed and $\alpha_3$ is variable, and we assume that $(\alpha_2\alpha_3,q)=1$. We show that for all $\alpha_3$ modulo $q$ which are coprime to $q$ except for a small number of $\alpha_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$ 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 Mili\'cevi\'c, replacing the exponent $11/24$ above by $11/25$. Under the Lindel\"of hypothesis for Dirichlet $L$-functions, we are able to replace the exponent $11/24$ above by $1/3$. Show Chinese abstract

Abstract: 我们考虑形如 $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$的二次同余的小解，其中$q=p^m$是奇素数幂。 这里，$\alpha_2$是任意但固定的，$\alpha_3$是变量，并且我们假设$(\alpha_2\alpha_3,q)=1$。 我们证明了对于所有与$q$互质的$\alpha_3$模$q$的情况（除了少数$\alpha_3$），当$q$趋向于无穷大时，如果$N\ge q^{11/24+\varepsilon}$成立，则关于同余式$x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$的解数$(x_1,x_2,x_3)$存在一个渐近公式，其中$\max\{|x_1|,|x_2|,|x_3|\}\le N$成立，并且该结论适用于所有奇素数幂的集合。 打破上述指数中的障碍1/2具有重要意义。 如果限制$q$为$p^m$的幂，其中$p$是{\it 固定的}素数，且$m$趋于无穷大时，我们利用 Milićević 发展的$p$-adic 指数对理论对此结果进行了轻微改进，将上述指数$11/24$替换为$11/25$。 在狄利克雷$L$-函数的林德勒夫假设下，我们可以将上面的指数$11/24$替换为$1/3$。