Gonality of the modular curve $X_0(N)$

Filip Najman; Petar Orlić

Gonality of the modular curve $X_0(N)$

Number Theory 2023-05-12 v4 Algebraic Geometry

Authors: Filip Najman , Petar Orlić

Abstract

In this paper we determine the $\mathbb Q$ -gonalities of the modular curves $X_0(N)$ for all $N<145$ . We determine the $\mathbb C$ -gonality of many of these curves and the $\mathbb Q$ -gonalities and $\mathbb C$ -gonalities for many larger values of $N$ . Using these results and some further work, we determine all the modular curves $X_0(N)$ of gonality $4$ , $5$ and $6$ over $\mathbb Q$ . We also find the first known instances of pentagonal curves $X_0(N)$ over $\mathbb C$ .

Keywords

modular curve x0(n) q-gonality c-gonality pentagonal curve modular form

Cite

@article{arxiv.2207.11650,
  title  = {Gonality of the modular curve $X_0(N)$},
  author = {Filip Najman and Petar Orlić},
  journal= {arXiv preprint arXiv:2207.11650},
  year   = {2023}
}

Comments

23 pages, this is the final version, to appear in Math. Comp

Gonality of the modular curve $X_0(N)$

Abstract

Keywords

Cite

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