发表机构
Department of Mathematics, Nanjing University（南京大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对非几何二次诱导表示，证明当权重对非有理时素数计数为亚多项式增长，当权重对为有理非整数时仅有有限多个，主要方法为刚性定理与Pila-Wilkie计数。
AI 中文摘要
设$K/\mathbb Q$为虚二次扩张，$p$为奇素数。记$\rho=\operatorname{Ind}_{G_K}^{G_{\mathbb Q}}\chi$，其中$E/\mathbb Q_p$为有限扩张，$\chi:G_K\to\mathcal O_E^\times$为连续特征。对固定的$r\in\mathbb Z\setminus\{0\}$，令$\pi_{\rho,r}(X)$表示满足$\ell\le X$、$\rho$在$\ell$处不分歧且$\operatorname{tr}\rho(\operatorname{Frob}_\ell)=r$的有理素数$\ell$的个数。设$a,b$为$\chi$在$p$处的两个权重。我们证明：若$(a,b)\notin\mathbb Q^2$，则对任意$\varepsilon>0$，有$\pi_{\rho,r}(X)\ll_{\rho,r,\varepsilon}X^\varepsilon$；而若$(a,b)\in\mathbb Q^2\setminus\mathbb Z^2$，则这样的素数仅有有限多个。这些界显著稀疏于经典CM Lang-Trotter尺度。非有理情形的主要输入是$p$-adic解析迹轨迹中代数曲线的刚性定理，结合刚性解析Pila-Wilkie计数；有理非整数情形则通过局部分歧论证处理。
英文摘要
Let $K/\mathbb Q$ be an imaginary quadratic extension and $p$ an odd prime. Write $ρ=\operatorname{Ind}_{G_K}^{G_{\mathbb Q}}χ$, where $E/\mathbb Q_p$ is a finite extension and $χ:G_K\to\mathcal O_E^\times$ is a continuous character. For a fixed $r\in\mathbb Z\setminus\{0\}$, let $π_{ρ,r}(X)$ denote the number of rational primes $\ell\le X$ such that $ρ$ is unramified at $\ell$ and $\operatorname{tr}ρ(\operatorname{Frob}_\ell)=r$. Let $a,b$ be the two weights of $χ$ at $p$. We prove that if $(a,b)\notin\mathbb Q^2$, then $π_{ρ,r}(X)\ll_{ρ,r,\varepsilon}X^\varepsilon$ for every $\varepsilon>0$, while if $(a,b)\in\mathbb Q^2\setminus\mathbb Z^2$, then only finitely many such primes occur. These bounds are substantially sparser than the classical CM Lang--Trotter scale. The main input in the non-rational case is a rigidity theorem for algebraic curves in the \(p\)-adic analytic trace locus, combined with rigid-analytic Pila--Wilkie counting; the rational non-integral case is treated by a local ramification argument.
