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

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.

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