Modular Curves X1(n) as Moduli of Point Arrangements

Explore a mathematical lecture on modular curves X1(n) as moduli of point arrangements delivered by Lev Borisov from Rutgers University at the Institute for Advanced Study's Special Year Seminar I. Learn about complex elliptic curves and how points of order n behave under Weierstrass embedding into CP2, with particular focus on when these points become collinear. Discover the proof that for n≥10, collections of n points in CP2 with specific collinearity properties generally derive from points of order n on elliptic curves. The talk covers interesting identities between logarithmic derivatives of theta functions at rational points and discusses potential applications to bounds on numbers of Hecke eigenforms for Γ1(n) of positive analytic rank. This research represents joint work with Xavier Roulleau, as documented in their arXiv paper 2404.04364.