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

Abstract

Let p be a prime number and let E/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.