**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.

thatsense, and no "hermitian" condition. Then I guess I don't know what condition(s) must be satisfied for a legitimate morphism "of Shimura varieties". The "hermitian condition" is about morphisms $G\to H$ over $\mathbb Q$ so that on non-compact factors over $\mathbb R$ the induced map on (hermitian) symmetric spaces $G_v/K_v$ is holomorphic. In that case, there is a "physical" induced map of complex analytic varieties... But this is evidently not what you're asking about! :) Thanks for clarifying. $\endgroup$1more comment