I saw the question: 


[Abelian varieties with CM](https://mathoverflow.net/questions/228310/abelian-varieties-with-cm)


and though I know that there are rare CM elliptic curves, I wonder 
what kind of curves with higher genus have the CM Jacobians?



Curves of higher genus
I am not sure whether this answers your question: it is a conjecture of Coleman that for a fixed genus $g$ sufficiently high, there should be only finitely many CM Jacobians of genus $g$. In fact Coleman stated the conjecture for $g\geq 4$, but by now there are counter-examples for $g\leq 7$ (at least). See for instance [this paper](https://www.math.uni-bielefeld.de/documenta/vol-15/24.pdf) of B. Moonen.


