Integral and p-adic Refinements of the Abelian Stark Conjecture

We give a formulation of the abelian case of Stark’s Main Conjecture in terms of determinants of projective modules and briefly show how this formulation leads naturally to its Equivariant Tamagawa Number Conjecture (ETNC) – type integral refinements. We discuss the Rubin-Stark integral refinement of an idempotent p1 iece of Stark’s Abelian Main Conjecture. In the process, we give a new formulation of its particular case, the Brumer-Stark conjecture, in terms of annihilators of generalized Arakelov class-groups (first Chow groups.) In this context, we discuss somewhat in detail our recent theorems with Greither, settling refinements of the Brumer-Stark conjecture, under certain hypotheses. We formulate a general Gross–type refinement of the Rubin–Stark conjecture, which has emerged fairly recently from work of Gross, Tate, Tan, Burns, Greither and the author, and interpret it in terms of special values of p–adic L–functions. Finally, we discuss the recent overwhelming evidence in support of the combined Gross-Rubin-Stark conjecture, which is a consequence of independent work of Burns and that of Greither and the author on the ETNC and various Equivariant Main Conjectures in Iwasawa theory, respectively.