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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 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.





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 X be a smooth algebraic variety over \mathbb{C}. A vector bundle on X is a locally free coherent (Zariski) sheaf \mathcal{F}, and a connection is an additive map of sheaves

\nabla: \mathcal{F} \rightarrow \mathcal{F} \otimes_{\mathcal{O}_X} \Omega^1_{X/\mathbb{C}} 

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


One can also make the exactly parallel construction over the complex manifold X^{an}. Given an algebraic differential equation, we can analytify it and obtain an analytic differential equation: (-)^{an}: DE(X) \rightarrow DE(X^{an}).

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

RH: DE(X^{an}) \rightarrow DE(X).

Theorem (Deligne): The functor RH (exists and) is fully faithful, and the essential image RSDE(X) is characterised by taking a good compactification of X 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 DE(X^{an}) than in DE(X) 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 \mathcal{F} \cong RH(\mathcal{F})^{an} but not (unless X is proper) an identification in the other direction.


Let’s see what happens in a stupid example. Let X=\mathbb{A}^1 and \mathcal{F} the trivial line bundle. Let’s take our favourite non-vanishing algebraic section v, and observe that any connection is automatically flat and determined by \nabla(v)

(since any other section can be written as f(t)v which by Leibniz is given by

\nabla(f(t)v) = v f'(t) \otimes dt + f(t) \nabla(v)).

If g(t) \in \mathbb{C}[t] let us denote by \nabla_g the connection taking v \mapsto g(t) v \otimes dt.


Let’s suppose we have a map \alpha: \nabla_g \rightarrow \nabla_h between two such differential equations. Let \alpha(v) = u(t) v, and the compatibility of \alpha with the connections gives the relation

u'(t) + u(t)h(t) = u(t)g(t).

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

u(t) = e^{\int (g(t)-h(t))dt}

which is never algebraic unless g(t)-h(t)=0.


The upshot is that in the algebraic category each \nabla_g gives a distinct differential equation, while in the analytic category one can use the above recipe to construct isomorphisms between each of the \nabla_g and in particular they are all equal to \nabla_0, which is the unique algebraic connection with regular singularities at \infty.