Gonality of the modular curve 𝑋₀(𝑁)

Najman, Filip; Orlić, Petar

doi:10.1090/mcom/3873

Gonality of the modular curve $X_0(N)$
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by Filip Najman and Petar Orlić

Math. Comp. 93 (2024), 863-886

DOI: https://doi.org/10.1090/mcom/3873

Published electronically: July 13, 2023

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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 $\mathbb {C}$.

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