Special Year Seminar I

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 $CP^2$ 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 $CP^2$  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 $\Gamma_1(n)$ of positive analytic rank, although this is rather speculative. 

This is joint work with Xavier Roulleau, see  https://arxiv.org/pdf/2404.04364.