  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-\'etale cohomology complexes' and of `integral Selmer
groups' for the multiplicative group $\mathbb{G}_m$ over finite abelian
extensions of number fields. We then conjecture a precise connection between
zeta elements and Weil-\'etale 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 $\mathbb{G}_m$, 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 $\mathbb{Q}$, a proof of new cases of the equivariant Tamagawa number
conjecture in situations in which the relevant $p$-adic $L$-functions have
trivial zeroes, explicit conjectural formulas for both annihilating elements
and, in certain cases, the higher Fitting ideals (and hence explicit
structures) of ideal class groups, a reinterpretation of the $p$-adic
Gross-Stark Conjecture in terms of the properties of zeta elements and a strong
refinement of many previous results (of several authors) concerning abelian
Stark conjectures.
