Intermediate modular curves with infinitely many quartic points

TL;DR

This work classifies intermediate modular curves $X_\Delta(N)$ that admit infinitely many quartic points over $\mathbb{Q}$ by developing a method to determine possible degrees of maps $X_\Delta(N)\to E$ to elliptic curves via a degree-pairing on $\operatorname{Hom}(J_\Delta(N),E)$. It combines conductor and Shimura-subgroup analyses with degeneracy maps to either produce a degree-4 morphism to $\mathbb{P}^1$ or a positive-rank elliptic curve, or to rule out such maps altogether. The authors provide an explicit finite list of $(N,\Delta)$ yielding infinite quartic points and prove nonexistence for the remaining cases, using both theoretical degree-form arguments and computational verification (Magna/Sage). The result advances understanding of low-degree points on modular curves and offers a transferable framework for examining higher-degree points via endomorphism and isogeny decompositions of Jacobians. The paper demonstrates how to combine modular-curve geometry, arithmetic of Jacobians, and computational tools to achieve a complete classification in this setting.

Abstract

For every group $\{\pm1\}\subseteq Δ\subseteq (\mathbb Z/N\mathbb Z)^\times$, there exists an intermediate modular curve $X_Δ(N)$. In this paper we determine all curves $X_Δ(N)$ with infinitely many points of degree $4$ over $\mathbb Q$. To do that, we developed a method to compute possible degrees of rational morphisms from $X_Δ(N)$ to an elliptic curve.

Theorems & Definitions (53)

• Theorem 1.1: Faltings' Theorem
• Theorem 1.2: Harris, Silverman: HarrisSilverman91
• Definition 1.3
• Theorem 1.4: Jeon: JEON2020272
• Theorem 1.5: Dalal: Dalal2023
• Theorem 1.6
• Proposition 1.7: Special case of \ref{['deltastrongweildegree']}
• Theorem 1.8: BELOV
• Proposition 1.9: DebarreFahlaoui
• Corollary 1.10