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Bielliptic and hyperelliptic modular curves $X(N)$ and the group $\mathrm {Aut}(X(N))$

Volume 161 / 2013

Francesc Bars, Aristides Kontogeorgis, Xavier Xarles Acta Arithmetica 161 (2013), 283-299 MSC: Primary 11G18; Secondary 11G30. DOI: 10.4064/aa161-3-6

Abstract

We determine all modular curves $X(N)$ (with $N\geq 7$) that are hyperelliptic or bielliptic. We also give a proof that the automorphism group of $X(N)$ is $\operatorname {PSL}_2(\mathbb {Z}/N\mathbb {Z})$, whence it coincides with the normalizer of $\varGamma (N)$ in $\operatorname {PSL}_2(\mathbb {R})$ modulo $\pm \varGamma (N)$.

Authors

  • Francesc BarsDepartament de Matemàtiques
    Universitat Autònoma de Barcelona
    08193 Bellaterra, Barcelona
    Catalonia, Spain
    e-mail
  • Aristides KontogeorgisK. Vesri 10 Papagou
    GR-15669, Athens, Greece
    e-mail
  • Xavier XarlesDepartament de Matemàtiques
    Universitat Autònoma de Barcelona
    08193 Bellaterra, Barcelona
    Catalonia, Spain
    e-mail

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