Modular Curves X1(n) as Moduli of Point Arrangements - Lev Borisov

Special Year Seminar I 2:00pm|Simonyi 101 Topic: Modular Curves X1(n) as Moduli of Point Arrangements Speaker: Lev Borisov Affiliation: Rutgers University Date: May 14, 2025 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. We prove that for n at least 10 a collection of n points in CP2 with these properties comes (generically) from a point of order n on an elliptic curve. In the process, we discover amusing identities between logarithmic derivatives of the theta function at rational points. I will also discuss potential applications of these results to bounds on the numbers of Hecke eigenforms for Γ1(n) of positive analytic rank, although this is rather speculative. This is joint work with Xavier Roulleau, see https://arxiv.org/pdf/2404.04364.