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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.

In this short post I want to note Deligne’s classic result on vector bundles with flat connections (let’s call them “differential equations”: terminology justified by the `cyclic vector theorem”) and give a stupid example illustrating the basic shape of things. I should thank Ananth Shankar for helping unconfuse me faster than I could unconfuse myself.

Let be a smooth algebraic variety over . A vector bundle on is a locally free coherent (Zariski) sheaf , and a connection is an additive map of sheaves

satisfying the Leibniz law. There is the usual notion of flatness of a connection. A morphism between two such objects is an -linear map between the underlying sheaves making the obvious square commute.

One can also make the exactly parallel construction over the complex manifold . Given an algebraic differential equation, we can analytify it and obtain an analytic differential equation:

Deligne’s result gives a functor in the other direction (let’s call it RH for “Riemann-Hilbert”)

.

**Theorem (Deligne): **The functor RH (exists and) is fully faithful, and the essential image is characterised by taking a good compactification of and restricting to those differential equations with regular singularities at the boundary.

This is a little counterintuitive at first. One imagines there being “more” objects in than in because one can write down more connections, but actually the extra freedom causes more stuff to become isomorphic, resulting in a “smaller” analytic category. Of course, one has but not (unless is proper) an identification in the other direction.

Let’s see what happens in a stupid example. Let and the trivial line bundle. Let’s take our favourite non-vanishing algebraic section , and observe that any connection is automatically flat and determined by

(since any other section can be written as which by Leibniz is given by

).

If let us denote by the connection taking .

Let’s suppose we have a map between two such differential equations. Let , and the compatibility of with the connections gives the relation

.

In other words, must satisfy a first order differential equation, and in fact we see that up to a constant factor

which is *never* algebraic unless .

The upshot is that in the algebraic category each gives a distinct differential equation, while in the analytic category one can use the above recipe to construct isomorphisms between each of the and in particular they are all equal to , which is the unique algebraic connection with regular singularities at .