# Shimura varieties of type C

Are there Shimura Varieties of Hodge-type which are not of PEL-type? I'd like to assume that the derived group is of type C.

• Yes --- PEL Shimura varieties are uncommon among those of Hodge type. Is the Shimura variety attached to $PGL_2$ of PEL-type?
– abz
Jan 29 '14 at 4:37
• @abz Thanks for pointing out my mistakes in my now deleted answer. I'd be glad to know a definitive answer to this question now. Jan 31 '14 at 9:38
• @abz thanks for your comment. What I was hoping to find out, though, was whether there exist Shimura Varieties of hodge type with no "extra endomorphisms" interpretation. From your answer, I'm guessing that there are plenty? Jan 31 '14 at 22:02

Without the restriction on the lie algebra, certainly. There are interesting examples of families of abelian varieties with (generically) no extra endomorphisms but whose generic Mumford-Tate group is not the whole of $GSp_{2g}$.
The idea is to take a division algebra $D$ over a cubic totally real field $F$ whose corestriction along $F/\mathbb{Q}$ is split, giving a natural map $Nm: D^* \rightarrow GL_8$. If $D$ ramifies at two of the three places at infinity, one shows that this map has symplectic image and can use the image to define a Hodge type Shimura datum. However, the representation is also absolutely irreducible which implies the underlying Hodge structure (hence abelian variety) attached to a general point in the family doesn't admit any extra endomorphisms.
However here I think the group is a $\mathbb{Q}$-form of $SO(4) \times SL(2)$, which maybe isn't what you are looking for (though I suppose its Lie algebra is of type $C_1=A_1$).
[P.S. to abz: unless I'm mistaken, $PGL_2$ only gives a Shimura variety of abelian type.]