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

Modular curves $X_1(n)$ 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 $p_k=kp$ 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 …