Modular curves X₁(n) as moduli spaces of point arrangements and applications

For a complex elliptic curve E and a point p of order n on it, the images of the points pₖ=kp under the Weierstrass embedding of E into CP² 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 C) to the modular curve X₁(n), provided n≥ 10, which also provides an integral model of X₁(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.