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.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2020
- DOI:
- 10.48550/arXiv.2004.13644
- arXiv:
- arXiv:2004.13644
- Bibcode:
- 2020arXiv200413644V
- Keywords:
-
- Mathematics - Number Theory;
- 11G16 (Primary) 14H52;
- 11G05;
- 14G35;
- 11F03 (Secondary)
- E-Print:
- 18 pages. An omission in the second proof of the main theorem has been corrected. Appendix B summarizes notation. Comments are welcome!