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Modular curves \(X_0(N)\) with infinitely many quartic points

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Abstract

We determine all modular curves \(X_0(N)\) with infinitely many quartic points. To do this, we define a pairing that induces a quadratic form representing all possible degrees of a rational morphism from \(X_0(N)\) to a positive rank elliptic curve.

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Data Availability Statement

The Sage codes used in the proofs can be found on: https://github.com/koffie/mdsage/tree/main/articles/derickx_orlic-quartic_X0.

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Acknowledgements

We are grateful to Filip Najman for his helpful comments and Kenneth A. Ribet for providing some very useful references to the literature. We also thank the referees for their comments that have greatly improved the paper.

Funding

The second author was supported by the QuantiXLie Centre of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004) and by the Croatian Science Foundation under the project no. IP-2022-10-5008.

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Authors and Affiliations

  1. Den Haag, The Netherlands

    Maarten Derickx

  2. University of Zagreb, Bijenička Cesta 30, 10000, Zagreb, Croatia

    Petar Orlić

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  1. Maarten Derickx
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  2. Petar Orlić
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Correspondence to Petar Orlić.

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Derickx, M., Orlić, P. Modular curves \(X_0(N)\) with infinitely many quartic points. Res. number theory 10, 42 (2024). https://doi.org/10.1007/s40993-024-00525-6

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