This question is somewhat confusing (and I was also confused by your previous question on Shimura curves). From reading this I am unsure whether you are interested in: a moduli of abelian threefolds? a Shimura variety parametrizing threefolds with extra structure (which is usually higher-dimensional than a curve)? Shimura curves (which usually parametrize abelian twofolds)? or functions from Shimura curves to the moduli of abelian threefolds? If you have something specific in mind, providing more information would make it easier for others to answer you.
– Tyler LawsonAug 5 '10 at 12:27

Isn't there are shimura variety of dimension one parametrizing abelian threefolds ? since Mumford has constructed a one dimension shimura vaiety parametrizing abelian fourfold [A Note of Shimura's paper"Discontinuous groups and abelian varieties"] ,so I wonder whether there are shimura varieties of dimension one parametrizing abelian threefold ,and what are they ?
– TOMAug 5 '10 at 13:00

or do you mean that :any shimura vaiety of one dimension ,if it parametrizing a family of abelian threefold ,then this family is a product of family of abelian surface with a fixed elliptic curve?
– TOMAug 5 '10 at 13:10

2

@TOM: I was thinking of PEL Shimura varieties parametrizing abelian threefolds, and the moduli of these is two-dimensional - but it's a different setup than the one Mumford uses. Perhaps someone else might know an appropriate group giving a curve? Re your 2. comment: It is not necessarily the case that such a family is a product (for instance, you might have a family which is isogenous to a product rather than isomorphic).
– Tyler LawsonAug 5 '10 at 16:17

1

Maybe I'm missing something, but I thought that the only reductive groups $/\mathbb{Q}$ arising in Shimura datum and such that the associated Hermitian symmetric space is of complex dimension 1 are forms of $GL_2$ associated to quaternion algebras : this dimension only depends on the semi-simple real lie group associated to $G^{ad}$, and one can compute dimensions in all the relevant cases (en.wikipedia.org/wiki/…). So in any case, Shimura curves are the "classical" ones (associated to forms of $GL_2$.
– Simon Pepin LehalleurAug 5 '10 at 17:45