In his note A Note of Shimura’s Paper “Discontinuous Groups and Abelian Varieties” Mumford constructs a Hodge type Shimura variety (a family of polarised abelian fourfolds with level structures) that is not of PEL type. He makes the necessary group from a quaternion algebra over a totally real field which is engineered to admit a symplectic representation which is absolutely irreducible (so the resulting abelian varieties cannot admit extra endomorphisms).

More specifically, let K/\mathbb{Q} be a cubic totally real field. We wish to carefully choose a quaternion algebra D/K such that H:= D^{N=1} admits a symplectic representation.

The trick is to note that corestriction on Brauer groups associates to D/K a quaternion algebra Cor(D)/\mathbb{Q}, equipped with a natural map Nm:D \rightarrow Cor(D). In particular, if we can arrange for Cor(D) to be split, we will certainly get a natural (8-dimensional) representation of D^*. By then imposing some conditions at infinity we can produce a symplectic structure on this representation, realising the resulting Shimura variety as one of Hodge type.

These conditions can be realised: just take D to be ramified at any even number of places above each place of \mathbb{Q}, and at two of the three places at infinity.

Then H(\mathbb{R}) = SU(2) \times SU(2) \times SL_2(\mathbb{R}), and unravelling the construction one sees that the map Nm: H(\mathbb{R}) \rightarrow SL_8(\mathbb{R}) factors SU(2) \times SU(2) \times SL(2) \rightarrow SO(4) \times SL(2) \rightarrow SL(8) where the first map is the standard degree 2 isogeny SU(2) \times SU(2) \rightarrow SO(4) and the second is given by the natural action on \mathbb{R}^4 \otimes \mathbb{R}^2, and visibly preserves the natural symplectic pairing coming from the product of orthogonal and symplectic pairings on these two spaces. Symplectic structures descend, giving the required map

H \rightarrow Sp_8  over \mathbb{Q}.

One extends this game to a Hodge type Shimura datum by taking 

G=Nm(D^*) \subset GL_{8,\mathbb{Q}}, which is an extension of the image of H by \mathbb{G}_m landing in GSp_8.

One takes the Hodge structure h to be the composite of Nm with the map taking a+ib \in \mathbb{S} to 1 on the GU(2)\times GU(2) factor of D^*_\mathbb{R} and the standard 2-dimensional embedding into the GL_2(\mathbb{R}) factor. It is obvious this extends to give the Siegel Shimura datum on GSp_8, so we have a Shimura datum of Hodge type, as required.

 

 

 

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