2014/10/17 by Abolfazl Mohajer, Mohajer, Abolfazl, Stefan Müller–Stach +3
Mathematics · #14G35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1410.4654
openalex publication_date 2014/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain D by a discrete subgroup. Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety Y in X (of Shimura type), and formulate necessary criteria for Y to be special. Our method consists in looking at finitely many compactified special curves Ci in Y, and testing whether the inclusion of the union of all Ci in Y satisfies certain properties. One of them is the so-called relative proportionality condition. In this paper, we give a new formulation of this numerical criterion in the case of Mumford-Tate varieties X. In this way, we give necessary and sufficient criteria for a subvariety Y of X to be a special subvariety in the sense of the Andre-Oort conjecture. We discuss in detail the important case where X=Ag, the moduli space of principally polarized abelian varieties.