October 16, 2025Open Access

Morphisms on the modular curve X₀ (p) and degree 6 points

Key Points

Morphisms from X0(p) to curves of genus ≥ 2 are classified, showing a strong relation with quotient maps.
For primes less than 3000, morphisms f:X0(p) to Y must be isomorphic to X0(p) to X0^+(p), reinforcing connections in algebraic geometry.
We have categorized all points of degree ≤ 25 on X0(p), highlighting the significance of modular curves in number theory.
The identification of curves X0(p) with infinitely many degree 6 points helps extend existing knowledge on small degree points.

Abstract

Let p be a prime. We study the morphisms f: X₀ (p) ₐ Y, where Y/ Q is a curve of genus 2. We prove that for p<3000 such an f must be isomorphic to the quotient map X₀ (p) X₀^+ (p). This allows us to classify all points of degree 25 on X₀ (p) that come from a map to some curve of genus 2. As an application, we were able to determine all curves X₀ (p) with infinitely many points of degree 6 over Q, continuing the previous results on small degree points on X₀ (N).

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Derickx et al. (Thu,) studied this question.

synapsesocial.com/papers/68f04acce559138a1a06e7a0 — DOI: https://doi.org/10.48550/arxiv.2506.21166

Authors

Maarten Derickx

University of Zagreb

Petar Orlić

University of Zagreb

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Morphisms on the modular curve X₀ (p) and degree 6 points

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Abstract

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Cite this study

Authors

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