This award continues support for the principal investigator's research on 'idempotent relations between mathematical objects'. This research generalizes and unifies the past work of Brauer on class numbers and of Accola on genera of Riemann surfaces. This research, set in the context of group schemes of finite type over the integers, explores the compatibility of certain conjectures such as those of Birch-Swinnerton-Dyer and of Tate having idempotent relations in group rings. This research is in the general area of number theory. Number theory is one of the oldest branches of research in the mathematical sciences. It concerns the basic structure of the integers, extensions of the integers, and the fundamental structure of the fields in which they lie. Research tools require a sophisticated blend of analysis, geometry, and algebra.
