Modular curves $X_1(n)$ as moduli spaces of point arrangements and applications

Abstract

For a complex elliptic curve $E$ and a point $p$ of order $n$ on it, the images of the points $p_k=kp$ under the Weierstrass embedding of $E$ into $\mathbb{C}\mathbb{P}^2$ are collinear if and only if the sum of indices is divisible by $n$. Thus, it provides a realization of a certain matroid. We study this matroid in detail and prove that its realization space is isomorphic (over $\mathbb{C}$) to the modular curve $X_1(n)$, provided $n\geq 10$, which also provides an integral model of $X_1(n)$. In the process, we find a connection to the classical Ceva and B\"or\"oczky examples of special point and line configurations. We also discuss the situation for smaller values of $n$.