Abstract
We study the splitting of primes in number fields generated by points on modular curves. Momose (Nagoya Math J 96:139–165, 1984) was the first to notice that quadratic points on \(X_1(N)\) generate quadratic fields over which certain primes split in a particular way and his results were later expanded upon by Krumm (Quadratic Points on Modular Curves. PhD thesis, University of Georgia, 2013). We prove results about the splitting behaviour of primes in quadratic fields generated by points on the modular curves \(X_0(N)\) which are hyperelliptic (except for \(N=37\)) and in cubic fields generated by points on \(X_1(2,14)\).
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Acknowledgements
We thank the referees and Andrej Dujella for many helpful comments that greatly improved the exposition of the paper.
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The authors were 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-2018-01-1313.
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Najman, F., Trbović, A. Splitting of primes in number fields generated by points on some modular curves. Res. number theory 8, 28 (2022). https://doi.org/10.1007/s40993-022-00325-w
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DOI: https://doi.org/10.1007/s40993-022-00325-w