First a disclaimer: I am at best a part-time arithmetic geometer, so please accept my apologies when I am too naive or get something wrong.

From time to time I have tried to learn something about Shimura varieties. The friendliest example I came across so far are Shimura curves arising from quaternion algebras. By this, I will mean the following:

Choose an indefinite quaternion algebra $B$ over $\mathbb{Q}$ with discriminant $D\neq 1$ and a maximal order $\Lambda \subset B$. Then $\mathcal{X}^B$ is the stack over $\mathbb{Z}[\frac1D]$ classifying, for $R$ an $\mathbb{Z}[\frac1D]$-algebra, projective abelian surfaces $A$ over $R$ with an action $\Lambda \to End(A)$ satisfying the following condition: If $R\to S$ is a ring map such that $\Lambda \otimes S \cong M_2(S)$, then the two summands of the $M_2(S)$-module $T_eA \otimes_R S$ are locally free of rank $1$.

These kinds of Shimura curves have (among others) the following nice properties:

- $\mathcal{X}^B$ is a smooth and proper Deligne-Mumford stack.
- The deformation theory is controlled by the associated $p$-divisible group. More precisely: We have an étale map $(\mathcal{X}^B)_p \to \mathcal{M}_{p-div}$ to the moduli stack of
*1-dimensional*$p$-divisible groups. - One has a good understanding of the coarse moduli space over $\mathbb{C}$ (quotient of upper half-plane).
- In many cases one can write down equations of coarse moduli spaces over $\mathbb{Q}$ or even $\mathbb{Z}$ of the curve or its quotient by the Atkin-Lehner involutions.
- Points (especially of maximal height of the $1$-dimensional formal group) can have interesting automorphism groups; but one can choose level structures so that points have only trivial automorphisms.
- Its definition does not require adèles. (This probably only an advantage due to my inexperience.)

Is there anything like this for Shimura varieties of dimension $2$? I do not need huge families, but only a few examples I can touch with my hands, with a reasonably simple definition and a good moduli interpretation (deformation theory controlled by $1$-dimensional $p$-divisible group).