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Title:Distribution of special points on modular varieties and
applications
Abstract: Let G be a connected reductive linear algebraic group over the field Q of rational numbers, K a maximal compact subgroup and \Gamma an arithmetic subgroup of G. Suppose that D=G(R)/K admits a G(R)-invariant complex structure and that V=\Gamma/D has the structure of a quasi-projective complex algebraic variety. The special points of V come from the fixed points in D of the maximal compact tori in G. These points play a key role in many arithmetic questions, especially when V is a modular variety for families of abelian varieties. We discuss some recent results about such points, in particular concerning automorphic functions and monodromy. Some of these results have their origins in Hilbert's 7th problem, in class field theory and in questions posed by Siegel on the exceptional sets of G=functions and modular functions. For example, if F=F(a,b,c,z) is the Gauss hypergeometric function (which is a G-function) and a, b, c are rational, the finiteness or not of certain sets of special values of F related to special points, determines the arithmetic nature of the monodromy group of F. |