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

2024/04/05 by Borisov, Lev, Roulleau, Xavier
#11Fxx #11G05 #14G35 #14K10 #14N20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2404.04364

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 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 Böröczky examples of special point and line configurations. We also discuss the situation for smaller values of n.

Related

• Modular Arrangements
• Chamber structure of modular curves X₁(N)
• Regular polygons, line operators, and elliptic modular surfaces as realization spaces of matroids
• Modular Compactifications of M_1,n II
• Modular curves and their pseudo-analytic cover