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 be a cubic totally real field. We wish to carefully choose a quaternion algebra such that admits a symplectic representation.

The trick is to note that corestriction on Brauer groups associates to a quaternion algebra , equipped with a natural map . In particular, if we can arrange for to be split, we will certainly get a natural (8-dimensional) representation of . 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 to be ramified at any even number of places above each place of , and at two of the three places at infinity.

Then , and unravelling the construction one sees that the map factors where the first map is the standard degree 2 isogeny and the second is given by the natural action on , 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

over .

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

, which is an extension of the image of by landing in .

One takes the Hodge structure to be the composite of with the map taking to on the factor of and the standard 2-dimensional embedding into the factor. It is obvious this extends to give the Siegel Shimura datum on , so we have a Shimura datum of Hodge type, as required.

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