An unexpected trace relation of CM points

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N=p^2M$ where $p$ is an odd prime not dividing $M$. Let $\mathcal{O}_f$ be the order of conductor $f$ (relatively prime to $N$) in an imaginary quadratic field $K$ in which $p$ is inert and such that the sign of the functional equation of $E/K$ is $-1$. Associated to these data there is a Shimura curve of non-split Cartan level at $p$ and a CM point of conductor $f$ on it. We can also consider a CM point of conductor $pf$ on another Shimura curve, using a split Cartan level at $p$. These curves admit parametrizations to $E$ and taking the images of the CM points we obtain points on $E$ defined over $H_f$ and $H_{pf}$ respectively (the ring class fields of conductor $f$ and $pf$). We prove that these points arising from different Shimura curves satisfy a trace compatibility that is non-trivial if and only if the local sign of $E/\mathbb{Q}$ at $p$ is $+1$.