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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.

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  • $\begingroup$ Yes --- PEL Shimura varieties are uncommon among those of Hodge type. Is the Shimura variety attached to $PGL_2$ of PEL-type? $\endgroup$ – abz Jan 29 '14 at 4:37
  • $\begingroup$ @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. $\endgroup$ – Olivier Jan 31 '14 at 9:38
  • $\begingroup$ @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? $\endgroup$ – user46141 Jan 31 '14 at 22:02
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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 $GSp_{2g}$.

I believe the first example, a family of fourfolds (the smallest dimension in which such a phenomenon arises) was given at the end of a short paper of Mumford "A note on Shimura's paper "Discontinuous groups and abelian varieties." http://dash.harvard.edu/bitstream/handle/1/3612771/McMullen_OnShimura.pdf?sequence=3

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.]

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