math.NT — Number Theory

Gonality and quintic points on intermediate modular curves

Contributed by Marin Varivoda

For $\{\pm 1\}\subseteq\Delta\subseteq(\mathbb{Z}/N\mathbb{Z})^\times$ let $X_\Delta(N)$ be the intermediate modular curve between $X_1(N)$ and $X_0(N)$. We determine the $\mathbb{Q}$-gonality of every strict intermediate curve $X_\Delta(N)$ with $N\le 50$, and of $X_1(N)$ for $41\le N\le 50$, where it equals the upper bound of Derickx and van Hoeij at every level. We also settle the thirty levels left open by Derickx, Hwang, Jeon and Orli\'c, completing the list of curves $X_0(N)$ with infinitely many points of degree $5$, and we show that exactly $43$ strict intermediate curves have infinitely many points of degree $5$. As a consequence, the strict intermediate curves of density degree $5$ are exactly the four $\mathbb{Q}$-pentagonal curves found by Orli\'c. The proofs use published classifications, the Castelnuovo--Severi inequality, and three computational methods; all computations can be rerun from the accompanying code.