# On the class of Shimura data of Hodge type that cover a given Shimura datum of abelian type

A Shimura datum $(G,X)$ is of Hodge type if there exists an injective morphism of Shimura data $(G,X) \hookrightarrow (\mathrm{GSp}_{2g}, \mathfrak{H}^{\pm})$, for some integer $g$.

Let $(G',X')$ be an adjoint Shimura datum; that is, $(G',X') = (G^{'\text{ad}},X^{'\text{ad}})$. Consider the class $\mathcal{C}$ consisting of all simple Shimura data of Hodge type $(G,X)$ for which $(G^{\text{ad}},X^{\text{ad}}) = (G',X')$. (The condition that $(G,X)$ is simple is there to exclude irrelevant torus factors coming from products with CM abelian varieties.)

By considering morphisms of Shimura data over $(G',X')$ we can endow $\mathcal{C}$ with a poset structure.

Assume that $(G',X')$ is of abelian type, which implies that $\mathcal{C}$ is not empty.

I would like to get a better feeling for $\mathcal{C}$. Here are a couple of concrete questions:

1. Are there bounds on the rank of $G$, for $(G,X)$ in $\mathcal{C}$?
2. Is the set of isomorphism classes in $\mathcal{C}$ finite?
3. Are there other interesting things to be said about the poset $\mathcal{C}$?
4. Is there (up to isomorphism) a canonical/natural Shimura datum $(G,X)$ in $\mathcal{C}$?

I do not have a good feeling for what kind of structure I should expect on $\mathcal{C}$. Maybe there is somehow a `minimal' element, or something like that. I have tried to put the classification by Deligne [section 1.3, Corvalis, 1979] to use; but I did manage to get something useful.

(After the first comment, by Keerthi Madapusi Pera, the question got a slight facelift. Initially I only asked the final question, about a natural Shimura datum $(G,X)$ in $\mathcal{C}$.)

• No, not in general. If you look at Deligne's construction, then he basically starts with the semi-simple cover of $G'$ that admits a minuscule representation of the right type (this cover can usually be taken to be simply connected), and then throws in a CM torus to fudge with the weights of the representation to make it look like the homology of an abelian variety. This 'fudging' torus is basically arbitrary, and there is no canonical choice. – Keerthi Madapusi Pera Mar 10 '17 at 16:38
• @KeerthiMadapusiPera That is a bit what I feared. But is it possible to put bounds on the rank of this 'fudging' torus? Of course you can always take products with CM abelian varieties, but that is a kind of cheating. So let us restrict to $(G,X)$ that are simple. I have no clue about the number of simple isomorphism classes in $\mathcal{C}$. I would not be surprised if it were finite, but I do not know what to expect. – user105976 Mar 11 '17 at 6:41