UNC-Duke Number Theory Seminar

"Cycles on Abelian surfaces"In this talk we use generalizations of classical geometric constructions of Kummer and Humbert to construct new higher Chow cycles on Abelian surfaces and K3 surfaces over p-adic local fields, generalising some work of Collino. The existence of these cycles is predicted by the poles of the local L-factor at p of the L-function of the Abelian surface. The techniques involve using some recent work of Bogomolov, Hassett and Tschinkel on the deformations of rational curves on K3 surfaces. As an application we use these cycles to prove an analogue of the Hodge-D-conjecture for Abelian surfaces.