Let be an algebraically closed field, , a polarized abelian variety. We are interested in:

- Field of moduli of and field of moduli of
- Fields of definition of and fields of definition of .

The field of moduli of is the unique minimal field in such that for all ).

. The top field is the unique field fixed by all such that there exists making the following diagram commute:

Shimura: the generic odd dimensional polarized abelian variety admits a model over . The generic even dimensional polarized abelian variety does not admit a model over .

(Silverberg) Fix . There is a (unique, Galois) minimal field of definition of .

(Silverberg) There is a such that for any abelian variety such that and with .

If is simple, is an order in either a totally real field, a division algebra over a CM-field, or a quaternion algebra.

We will focus on the latter case.

*Forgetful maps between Shimura varieties and rational points*

Let be a totally real number field, with . Let be an indefinite quaternion algebra over (that is, ). Let be a maximal order in . Let

Moduli problem : classify principally polarized abelian varieties where is an abelian variety of , ,and the Rosati involution has the form , where such that and .

(Shimura): The moduli functor is coarsely represented over by a complete algebraic variety , with . We havea , where is the Poincaré upper half plane.

Let be the ring of integers of , , where is a totally real quadratic order over . There are forgetful finite maps over

where is the Hilbert modular variety classifying varieties with real multiplication by the subscript. The dimensions of these moduli spaces are, respectively, , , , .

The picture in is

Shimura curve: Hilbert surface: Igusa's space: |

We have a tower of fields

The automorphism group of : is

where .

*Field of definition for abelian surfaces*

Let be a principally polarized abelian surface over a number field with , . There is a diagram

- If
for ,
, then (with Dieulefait)

- Otherwise,

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