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

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The theory of special values of L-functions is a major, active area ofresearch within the general fields of number theory and arithmeticalgebraic geometry. Stark's Main Conjecture provides a link betweenspecial values of Artin L-functions and the arithmetic of the associatedGalois extensions. In recent years, Rubin and Popescu have formulatedrefined, integral versions of Stark's Main Conjecture in the case ofabelian L-functions of arbitrary order of vanishing at the origin. Also,Burns and Flach, by reworking earlier conjectures of Bloch-Kato andFontaine-Perrin Riou, have formulated the Equivariant Tamagawa NumberConjecture for certain classes of motivic L-functions. If restricted tothe case of Artin L-functions, the Burns-Flach conjecture can also beviewed as a refined, integral version of Stark's Main Conjecture. ThePrincipal Investigator focuses on providing evidence for, studying thefunctorial behavior of, and finding links between the Conjectures ofRubin, Popescu, and Burns-Flach. He also works on developingGross-type p-adic refinements of these statements, as well as buildingbridges between these statements and the Theory of Euler Systems,Equivariant Iwasawa Theory, and the Conjectures of Brumer, Leopoldt, andChinburg.The L-functions are mathematical objects of analytic (continuous) nature,encoding an enormous amount of extremely interesting and usefulinformation of arithmetic (discrete) nature. The main goal of thisproject is to continue a program initiated by Stark, Rubin, the principalinvestigator, and Burns-Flach, and develop general recipes (conjectures)aimed at retrieving the arithmetic data encoded in a special type ofL-functions (the Artin L-functions), and follow these recipes (in otherwords prove these conjectures) in several important special cases. Inparallel, the Principal Investigator is developing links between theseconjectures and other, already developed theories, dealing with objects ofarithmetic (discrete) nature, such as the theory of Euler Systems andEquivariant Iwasawa Theory. Aside from its importance for the centralareas of pure mathematics called number theory and arithmetic algebraicgeometry, this research could have far reaching practical applications tothe development of new data encryption algorithms.