A Lang-Trotter Problem for Non-Geometric Quadratic Inductions

Let $K\mathrm{/}\mathbb{Q}$ be an imaginary quadratic extension and $p$ an odd prime. Write $\rho ={\mathrm{Ind}}_{G_K}^{G_{\mathbb{Q}}}\chi$, where $E\mathrm{/}{\mathbb{Q}}_p$ 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 $\mathrm{\ell }\le X$ such that $\rho$ is unramified at $\mathrm{\ell }$ and $\mathrm{tr}\rho ({\mathrm{Frob}}_{\mathrm{\ell }})=r$. Let $a,b$ be the two weights of $\chi$ at $p$. We prove that if $(a,b)\mathrm{\notin }{\mathbb{Q}}^2$, 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.