On Zeta Elements for G_m

In this paper, we present a unifying approach to the general theory of
abelian Stark conjectures. To do so we define natural notions of `zeta
element', of `Weil-étale cohomology complexes' and of `integral Selmer
groups' for the multiplicative group $\GG_{m}$ over finite abelian extensions
of number fields. We then conjecture a precise connection between zeta
elements and Weil-étale cohomology complexes, we show this conjecture
is equivalent to a special case of the equivariant Tamagawa number conjecture
and we give an unconditional proof of the analogous statement for global
function fields. We also show that the conjecture entails much detailed
information about the arithmetic properties of generalized Stark elements
including a new family of integral congruence relations between Rubin-Stark
elements (that refines recent conjectures of Mazur and Rubin and of the
third author) and explicit formulas in terms of these elements for the
higher Fitting ideals of the integral Selmer groups of **G**_{m},
thereby obtaining a clear and very general approach to the theory of abelian
Stark conjectures. As first applications of this approach, we derive, amongst
other things, a proof of (a refinement of) a conjecture of Darmon concerning
cyclotomic units, a proof of (a refinement of) Gross's `Conjecture for
Tori' in the case that the base field is Q, explicit conjectural formulas
for both annihilating elements and, in certain cases, the higher Fitting
ideals (and hence explicit structures) of ideal class groups and a strong
refinement of many previous results concerning abelian Stark conjectures.

2010 Mathematics Subject Classification:

Keywords and Phrases:

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