Modular curves of prime-power level with infinitely many quadratic points

Abstract

We completely determine the $1085$ open subgroups $H$ of ${\mathrm{GL}}_2(\hat{\mathbb{Z}})$ of prime-power level that satisfy $-I\in H$ and $\det (H)={\hat{\mathbb{Z}}}^{\times }$ for which the corresponding modular curve $X_H$ has infinitely many quadratic points. When $g(X_H)\ge 2$ this is equivalent to determining all the hyperelliptic modular curves of prime-power level and all the bielliptic modular curves of prime-power level that admit a degree two map to a positive rank elliptic curve. From the moduli perspective, this means that there are exactly 1085 subgroups $H$ of ${\mathrm{GL}}_2(\hat{\mathbb{Z}})$ of prime-power level for which there are infinitely many elliptic curves $E/K$ over quadratic extensions such that $\ddot{I}{}_E(G_k)$ is conjugate to a subgroup of $H$.

26 pages. Exposition changed after referee report. Comments welcome