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arXiv · 2509.22895

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

Michael Cerchia·Rakvi
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Abstract

We completely determine the $1085$ open subgroups $H$ of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ of prime-power level that satisfy $-I \in H$ and $\operatorname{det}(H)=\widehat{\mathbb{Z}}^{\times}$ for which the corresponding modular curve $X_H$ has infinitely many quadratic points. When $g(X_H)\geq 2$ 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 $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ of prime-power level for which there are infinitely many elliptic curves $E/K$ over quadratic extensions such that $\rho_E(G_k)$ is conjugate to a subgroup of $H$.

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Michael Cerchia, Rakvi. 2025-09-26. Modular curves of prime-power level with infinitely many quadratic points. https://arxiv.org/abs/2509.22895

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