**Question:** Let $S$ be a 0-dimensional Shimura variety. Does $S$ necessarily admit a morphism (in the category of Shimura varieties) to $\mathcal{A}\_g$ for some $g\geq 1$? Here $\mathcal{A}\_g$ is the moduli space of principally polarized abelian varieties.


**Motivation:** Basically, I am trying to understand something for all Shimura varieties. This reduces to some statement about 0-dimensional Shimura varieties, and I wanna see if I can apply techniques from the theory of Abelian Varieties to do so. 



Do all 0-dimensional Shimura Varieties show up (as CM points) in $\mathcal{A}_g$?
You can always map $(T,h)$ to the trivial Shimura datum and then this into the Siegel one. I assume this isn't what you want however. May I therefore modify the question to ask whether a zero dimensional variety can be *embedded* in a Siegel variety. I.e. You want to know if all zero dimensional Shimura varieties are of Hodge type. 


I think this is certainly false and can give two quick reasons.


1) every Hodge type datum has weight defined over $\mathbb{Q}$ and has compact centre modulo the image of this weight. In general neither condition need hold for a zero dimensional Shimura datum.


2) consider the 'cyclotomic' Shimura datum $T=\mathbb{G}\_m$ and $h(z) = z\bar{z}$. Any representation of T will induce a hodge structure of even weight but if it admits a Siegel embedding then this must yield a representation giving the weight 1 Hodge structure defining an abelian variety. 


