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Fields of definition of elliptic curves with prescribed torsion

Volume 181 / 2017

Peter Bruin, Filip Najman Acta Arithmetica 181 (2017), 85-95 MSC: Primary 11G05; Secondary 11G18. DOI: 10.4064/aa170323-20-9 Published online: 27 October 2017

Abstract

We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\mathbb{Q}$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.

Authors

  • Peter BruinMathematisch Instituut
    Universiteit Leiden
    Postbus 9512
    2300 RA Leiden, Netherlands
    e-mail
  • Filip NajmanDepartment of Mathematics
    University of Zagreb
    Bijenička cesta 30
    10000 Zagreb, Croatia
    e-mail

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