  We show that the l-adic realizations of certain Picard 1-motives associated
to a G-Galois cover of smooth, projective curves defined over an algebraically
closed field are G-cohomologically trivial, for all primes l. In the process,
we generalize a well-known theorem of Nakajima on the Galois module structure
of certain spaces of Kahler differentials associated to the top curve of the
cover, assuming that the field of definition is of characteristic p. If the
cover and the Picard 1-motive are defined over a finite field and if G is
abelian, we compute the first Fitting ideal of each l-adic realization over a
certain profinite l-adic group algebra in terms of an equivariant Artin
L-function. This is a refinement of earlier work of Deligne and Tate on Picard
1-motives associated to global function fields. As a consequence, we prove
refined versions of the Brumer-Stark and the (\'etale) Coates-Sinnott
Conjectures for global function fields. In upcoming work, we use these results
to prove several other conjectures on special values of global L-functions in
characteristic p. Also, we prove analogous results in characteristic 0, where
the l-adic realizations of Picard 1-motives will be replaced by a new class of
Iwasawa modules.

Number Theory;Algebraic Geometry