Stark-type Conjectures "over Z" and the Equivariant Tamagawa Number Conjecture

University Of California-San Diego, La Jolla CA

Investigators

Abstract

The theory of special values of L-functions is a major, active area of research within the general fields of number theory and arithmetic algebraic geometry. Stark's Main Conjecture provides a link between special values of Artin L-functions and the arithmetic of the associated Galois extensions. In recent years, Rubin and Popescu have formulated refined, integral versions of Stark's Main Conjecture in the case of abelian L-functions of arbitrary order of vanishing at the origin. Also, Burns and Flach, by reworking earlier conjectures of Bloch-Kato and Fontaine-Perrin Riou, have formulated the Equivariant Tamagawa Number Conjecture for certain classes of motivic L-functions. If restricted to the case of Artin L-functions, the Burns-Flach conjecture can also be viewed as a refined, integral version of Stark's Main Conjecture. The Principal Investigator focuses on providing evidence for, studying the functorial behavior of, and finding links between the Conjectures of Rubin, Popescu, and Burns-Flach. He also works on developing Gross-type p-adic refinements of these statements, as well as building bridges between these statements and the Theory of Euler Systems, Equivariant Iwasawa Theory, and the Conjectures of Brumer, Leopoldt, and Chinburg. The L-functions are mathematical objects of analytic (continuous) nature, encoding an enormous amount of extremely interesting and useful information of arithmetic (discrete) nature. The main goal of this project is to continue a program initiated by Stark, Rubin, the principal investigator, and Burns-Flach, and develop general recipes (conjectures) aimed at retrieving the arithmetic data encoded in a special type of L-functions (the Artin L-functions), and follow these recipes (in other words prove these conjectures) in several important special cases. In parallel, the Principal Investigator is developing links between these conjectures and other, already developed theories, dealing with objects of arithmetic (discrete) nature, such as the theory of Euler Systems and Equivariant Iwasawa Theory. Aside from its importance for the central areas of pure mathematics called number theory and arithmetic algebraic geometry, this research could have far reaching practical applications to the development of new data encryption algorithms.