Heegner points on Cartan non-split curves

Abstract

Let $E$ be an elliptic curve of conductor $N$, and let $K$ be an imaginary quadratic field such that the root number of $E/K$ is $-1$. Let $O$ be an order in $K$ and assume that there exists an odd prime $p$, such that $p^2\mid \mid N$, and $p$ is inert in $O$. Although there are no Heegner points on $X_0(N)$ attached to $O$, in this article we construct such points on Cartan non-split curves. In order to do that we give a method to compute Fourier expansions for forms in Cartan non-split curves, and prove that the constructed points form a Heegner system as in the classical case.

25 pages, revised version

References in corpus (1)

• Euler factors determine local Weil representations

Cited by in corpus (2)

• An unexpected trace relation of CM points
• A Gross-Kohnen-Zagier theorem for non-split Cartan curves