Transcendence on Varieties in Families

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

We propose applying transcendence techniques to curves and abelian varieties in a family and to classical period maps to deduce transcendence properties for functions which are solutions of a Picard-Fuchs equation. This leads naturally to considering the distribution of points with special properties on subvarieties of Shimura varieties. These special properties can be interpreted as meaning that the points lie on the intersection of the subvariety with certain Shimura subvarieties. This leads to studying open conjectures in the theory of Shimura varieties. We also propose an approach independent of these conjectures which would involve studying the locus of ``irreducible'' points on subvarieties of Shimura varieties. We propose developing a transcendence theory for more general varieties in a family, and for more general period maps as well as for periods of higher order forms. A fundamental problem in transcendence theory is the determination of the exceptional set of a function, that is the set of algebraic numbers at which the function assumes algebraic values. A classical result is that the exceptional set of the exponential function consists only of the origin. This implies that e and pi are transcendental. In previous work, we determine the exceptional set of the classical hypergemetric function and its generalization to several complex variables by relating these functions to families of algebraic curves. We intend to extend this work to a wider class of functions related to more general families of algebraic curves and varieties.