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

## TL;DR

This paper extends the Gross-Kohnen-Zagier theorem to non-split Cartan curves, linking special points on elliptic curves with Fourier coefficients of specific Jacobi forms, revealing new arithmetic relations.

## Contribution

It establishes a new Gross-Kohnen-Zagier type relation for elliptic curves with conductor p^2 and odd rank, involving non-split Cartan curves and Jacobi forms.

## Key findings

- Positions of special points are encoded in Fourier coefficients of a Jacobi form.
- The result generalizes classical Gross-Kohnen-Zagier theorem to non-split Cartan settings.
- Provides new insights into the arithmetic of elliptic curves with specific conductors.

## 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.