A Gross-Kohnen-Zagier theorem for non-split Cartan curves

Abstract

Let $p$ be a prime number and let $E/\mathbb{Q}$ be an elliptic curve of conductor $p^2$ and odd analytic rank. We prove that the positions of its special points arising from non-split Cartan curves and imaginary quadratic fields where $p$ is inert are encoded in the Fourier coefficients of a Jacobi form of weight $6$ and lattice index of rank $9$, obtaining a result analogous to that of Gross, Kohnen and Zagier.