I am trying to understand CM points on Shimura curves and I got confused. Before I get the point that I got confused and stuck I need to introduce some notations.
$F$: a number field, $\mathbb{A}_f$: finite adeles of $F$, $B$: quaternion algebra over $F$, $\mathcal{O}_B$: maximal order of $B$, $K$ : fixed imaginary quadratic extension of $F$ embedded into $B$.
$G$: algebraic group $B^*$ over $F$, $U$ :maximal compact subgroup of $G(\mathbb{A}_f)$.
Then we have a Shimura curve as the double coset space
$$\mathbb{S}_U = G(F) \backslash \mathbb{H} \times G(\mathbb{A}_f)/U$$
where here $\mathbb{H}$ is the upper half plane.
And we know that if $F=\mathbb{Q}$ then $\mathbb{S}_U$ parametrizes abelian surfaces.
Now I saw two different definitions of CM points on $\mathbb{S}_U$:
- By using moduli interpretion: A point $x \in \mathbb{S}_U$ is a CM point by K if the corresponding abelian variety has CM by K
- A point $x \in \mathbb{S}_U$ is a CM point by K if it is represented by $(z,g) \in \mathbb{H} \times G(\mathbb{A}_f)$ with $z$ fixed by a torus of $G(F)$ isomorphic to $K^*/\mathbb{Q}^*$.
I think these two definitions should coincide but I don't see any relation between these two. I tried considering an abelian variety with CM and try to get this fixed point of torus but I couldn't. And I couldn't find a source explaining this also. Any hint, source, idea, explanation would be very helpful. Thank you!