Let be an algebraically closed field, , a polarized abelian variety. We are interested in:
The field of moduli of is the unique minimal field in such that for all ).
(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
|Shimura curve: Hilbert surface: Igusa's space:|
The automorphism group of : is
Field of definition for abelian surfaces
Let be a principally polarized abelian surface over a number field with , . There is a diagram
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