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This is a Quantitative Study study.

October 1, 2025Open Access

Intermediate modular curves with infinitely many quartic points

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All curves XΔ(N) with infinitely many degree 4 points over Q are determined, establishing connections between modular curves and elliptic curves.
The developed method computes possible degrees of rational morphisms from XΔ(N) to elliptic curves, essential for understanding these curves' structure.
This study uncovers a relationship between modular curves and their quartic points, contributing to the field of algebraic geometry and number theory.
Findings enhance knowledge about rational morphisms and their implications in the context of modular curves and elliptic curves.

Implication

This research identifies modular curves with infinitely many degree 4 points over Q, highlighting methods for rational morphisms to elliptic curves.

Abstract

For every group \±1\⊆ Δ⊆ ( Z/N Z)^×, 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 Q. To do that, we developed a method to compute possible degrees of rational morphisms from X_Δ(N) to an elliptic curve.

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Cite This Study

Derickx et al. (2025) studied this question.

synapsesocial.com/papers/68dd91cbfe798ba2fc498735 https://doi.org/10.48550/arxiv.2504.15937

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Implication

This research identifies modular curves with infinitely many degree 4 points over Q, highlighting methods for rational morphisms to elliptic curves.

Intermediate modular curves with infinitely many quartic points

Key Points

Implication

Abstract

Discussion

Cite This Study

Intermediate modular curves with infinitely many quartic points

Key Points

Implication

Abstract

Discussion

Cite This Study

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