Geodesic
Bielliptic and hyperelliptic modular curves $X(N)$ and the group $\mathrm {Aut}(X(N))$
Francesc Bars 1   ; Aristides Kontogeorgis 2   ; Xavier Xarles 1
1 Departament de Matemàtiques Universitat Autònoma de Barcelona 08193 Bellaterra, Barcelona Catalonia, Spain
2 K. Vesri 10 Papagou GR-15669, Athens, Greece
Acta Arithmetica, Tome 161 (2013) no. 3, pp. 283-299 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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 $\pm \varGamma (N)$.
DOI : 10.4064/aa161-3-6
Keywords: determine modular curves geq hyperelliptic bielliptic proof automorphism group operatorname psl mathbb mathbb whence coincides normalizer vargamma operatorname psl mathbb modulo vargamma

Francesc Bars  1   ; Aristides Kontogeorgis  2   ; Xavier Xarles  1

1 Departament de Matemàtiques Universitat Autònoma de Barcelona 08193 Bellaterra, Barcelona Catalonia, Spain
2 K. Vesri 10 Papagou GR-15669, Athens, Greece 
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     author = {Francesc Bars and Aristides Kontogeorgis and Xavier Xarles},
     title = {Bielliptic and hyperelliptic modular curves $X(N)$
 and the group $\mathrm {Aut}(X(N))$},
     journal = {Acta Arithmetica},
     pages = {283--299},
     year = {2013},
     volume = {161},
     number = {3},
     doi = {10.4064/aa161-3-6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4064/aa161-3-6/}
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 and the group $\mathrm {Aut}(X(N))$
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Francesc Bars; Aristides Kontogeorgis; Xavier Xarles. Bielliptic and hyperelliptic modular curves $X(N)$
 and the group $\mathrm {Aut}(X(N))$. Acta Arithmetica, Tome 161 (2013) no. 3, pp. 283-299. doi: 10.4064/aa161-3-6

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