A Lang-Trotter Problem for Non-Geometric Quadratic Inductions

Abstract

Let $K/\mathbb Q$ be an imaginary quadratic extension and $p$ an odd prime. Write $\rho=\operatorname{Ind}_{G_K}^{G_{\mathbb Q}}\chi$, where $E/\mathbb Q_p$ is a finite extension and $\chi:G_K\to\mathcal O_E^\times$ is a continuous character. For a fixed $r\in\mathbb Z\setminus\{0\}$, let $\pi_{\rho,r}(X)$ denote the number of rational primes $\ell\le X$ such that $\rho$ is unramified at $\ell$ and $\operatorname{tr}\rho(\operatorname{Frob}_\ell)=r$. Let $a,b$ be the two weights of $\chi$ at $p$. We prove that if $(a,b)\notin\mathbb Q^2$, then $\pi_{\rho,r}(X)\ll_{\rho,r,\varepsilon}X^\varepsilon$ 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.