Number Theory and Allied Topics

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Abstract for award DMS-0355390 of BrownawellThis project focuses primarily on establishing function fieldanalogues of major conjectures in transcendence theory which arecurrently far out of the reach of methods in the classical setting ofthe complex numbers. The first step (joint with Papanikolas) will bethe analogue of the converse of the Shimura-Deligne relations on theperiods of abelian varieties. This will extend joint work withAnderson and Papanikolas precisely determining the linear relations onmonomials in special Gamma values. Other topics will involveindependence of divided derivatives of periods, transcendence andvariation of characteristic through powers of a fixed prime, andinvestigation of applications of interpolation matrices. Anotherdirection will be to approach Nesterenko's theorem for Ramanujan'sfunctions based on the investigator's independence criterion. Lastly,a ``final'' version of the arithmetic Liouville-Lojasiewicz inequalityis planned.As has been understood since the middle of the nineteenth century,rational functions in one variable behave amazingly like the rationalnumbers. Questions in this setting are fascinating in their ownright, and often they are easier to resolve than their classicalanalogues. This project pursues an extension in several well-definedsettings in function fields of the basic premise of all transcendenceinvestigations since Hermite: There are no surprising algebraicrelationships. By that we mean that all algebraic relationshipsbetween numbers that one might care are consequences of some extremelybasic underlying structure.