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 $\rho ={\mathrm{Ind}}_{G_K}^{G_{\mathbb{Q}}}\chi$, 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 $\mathrm{tr}\rho ({\mathrm{Frob}}_{\ell })=r$. Let $a,b$ be the two weights of $\chi$ at $p$. We prove that if $(a,b)\notin {\mathbb{Q}}^2$, then ${\pi }_{\rho ,r}(X){\ll }_{\rho ,r,\epsilon }{X}^{\epsilon }$ for every $\epsilon >0$, 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.