Number Theory and Allied Topics

Pennsylvania State Univ University Park, University Park PA

Investigators

Abstract

Abstract for award DMS-0355390 of Brownawell This project focuses primarily on establishing function field analogues of major conjectures in transcendence theory which are currently far out of the reach of methods in the classical setting of the complex numbers. The first step (joint with Papanikolas) will be the analogue of the converse of the Shimura-Deligne relations on the periods of abelian varieties. This will extend joint work with Anderson and Papanikolas precisely determining the linear relations on monomials in special Gamma values. Other topics will involve independence of divided derivatives of periods, transcendence and variation of characteristic through powers of a fixed prime, and investigation of applications of interpolation matrices. Another direction will be to approach Nesterenko's theorem for Ramanujan's functions based on the investigator's independence criterion. Lastly, a ``final'' version of the arithmetic Liouville-Lojasiewicz inequality is planned. As has been understood since the middle of the nineteenth century, rational functions in one variable behave amazingly like the rational numbers. Questions in this setting are fascinating in their own right, and often they are easier to resolve than their classical analogues. This project pursues an extension in several well-defined settings in function fields of the basic premise of all transcendence investigations since Hermite: There are no surprising algebraic relationships. By that we mean that all algebraic relationships between numbers that one might care are consequences of some extremely basic underlying structure.