Canonical Models and Special Points
Each Shimura variety can be defined over a canonical number field E called the reflex field. This important result due to Shimura shows that Shimura varieties, which a priori are only complex manifolds, have an algebraic field of definition and, therefore, arithmetical significance. It forms the starting point in his formulation of the reciprocity law, where an important role is played by certain arithmetically defined special points.
The qualitative nature of the Zariski closure of sets of special points on a Shimura variety is described by the André-Oort conjecture. Conditional results have been obtained on this conjecture, assuming a Generalized Riemann Hypothesis.
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