Mathematical Sciences: Problems in Number Theory

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This research concerns idempotent relations between mathematical objects. It generalizes and unifies the work of R. Brauer on class numbers and R. Accola on genera of Riemann surfaces. The research will relate these idempotent relations to the quantities involved in the Birch-Swinnerton-Dyer conjecture for abelian varieties over number fields. The research is in the general area of algebraic number theory. It concerns geometric transformation groups and their subgroups acting on mathematical objects. These transformation group actions determine numerical invariants and, in particular, idempotent relations. In this context, the researchers link geometric objects on one hand and arithmetic objects on the other hand. These links provide insights into both the geometry and the number theory.