Background:

Given an elliptic curve $E$, it seems that $max(ord(Aut(E)))$ over the prime 2 is 24, and $(max(ord(Aut(E)))$ over the prime 3 is 12.

The endomorphism algebra of an elliptic curve over $\overline{\mathbb{F}}_p$ is, after tensoring with $\mathbb{Q}$, either an imaginary quadratic field or the unique division algebra $D_p$ which is non-split at $p$ and infinity. So, in order to compute the maximal order of the automorphism algebra, we just compute the units of $D_p$, and then count them.

Here's my question: Given a variety, how does one compute the maximal order of its automorphism group? I'm particularly interested in the cases of abelian varieties with complex multiplication, and Shimura varieties.

For example, the variety $\mathbb{C}^{\times 3}/ \mathbb{Z}[\zeta_7]^{\times 3}$ where the embedding is given by $\sigma_1 \times \sigma_2 \times \sigma_3: \mathbb{Z}[\zeta_7]^{\times 3} \to \mathbb{C}^{\times 3}$.

I have heard that there is a theorem that $Aut(A, p)$ is finite, where $p$ is a polarization of the abelian variety $A$. Is there a general method to calculate this as well?

Background:

Given an elliptic curve $E$, it seems that $max(ord(Aut(E)))$ over the prime 2 is 24, and $(max(ord(Aut(E)))$ over the prime 3 is 12.

The endomorphism algebra of an elliptic curve over $\overline{\mathbb{F}}_p$ is, after tensoring with $\mathbb{Q}$, either an imaginary quadratic field or the unique division algebra $D_p$ which is non-split at $p$ and infinity. So, in order to compute the maximal order of the automorphism algebra, we just compute the units of $D_p$, and then count them.

Here's my question: Given a Shimura variety, how does one compute the maximal order of its automorphism group?

For example, the variety $\mathbb{C}^{\times 3}/ \mathbb{Z}[\zeta_7]^{\times 3}$ where the embedding is given by $\sigma_1 \times \sigma_2 \times \sigma_3: \mathbb{Z}[\zeta_7]^{\times 3} \to \mathbb{C}^{\times 3}$. Here, we look at its automorphisms over $\mathbb{F}_p$ where $p$ is $2$ or $4$ mod $7$.

I have heard that Shimura varieties have finite dimensional automorphism groups (analogous to the fact that $Aut(A, p)$ is finite, where $p$ is a polarization of the abelian variety $A$). What is the automorphism group of this Shimura variety?