A Divisor Formula and a Bound on the $\mathbb{Q}$-gonality of the Modular Curve $X_1(N)$

Mark van Hoeij; Hanson Smith

A Divisor Formula and a Bound on the $\mathbb{Q}$-gonality of the Modular Curve $X_1(N)$

Number Theory 2020-05-05 v2

Authors: Mark van Hoeij , Hanson Smith

Abstract

We give a formula for divisors of modular units on $X_1(N)$ and use it to prove that the $\mathbb{Q}$ -gonality of the modular curve $X_1(N)$ is bounded above by $\left[\frac{11N^2}{840}\right]$ , where $[\bullet]$ denotes the nearest integer.

Keywords

modular curve modular units divisor formula q-gonality bounded upper bound

Cite

@article{arxiv.2004.13644,
  title  = {A Divisor Formula and a Bound on the $\mathbb{Q}$-gonality of the Modular Curve $X_1(N)$},
  author = {Mark van Hoeij and Hanson Smith},
  journal= {arXiv preprint arXiv:2004.13644},
  year   = {2020}
}

Comments

18 pages. An omission in the second proof of the main theorem has been corrected. Appendix B summarizes notation. Comments are welcome!

A Divisor Formula and a Bound on the $\mathbb{Q}$-gonality of the Modular Curve $X_1(N)$

Abstract

Keywords

Cite

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