  In this paper, we study the Iwasawa theory of a motive whose Hodge-Tate
weights are $0$ or $1$ (thence in practice, of a motive associated to an
abelian variety) at a non-ordinary prime, over the cyclotomic tower of a number
field that is either totally real or CM. In particular, under certain technical
assumptions, we construct Sprung-type Coleman maps on the local Iwasawa
cohomology groups and use them to define integral $p$-adic $L$-functions and
(one unconditionally and other conjecturally) cotorsion Selmer groups. This
allows us to reformulate Perrin-Riou's main conjecture in terms of these
objects, in the same fashion as Kobayashi's $\pm$-Iwasawa theory for
supersingular elliptic curves. By the aid of our theory of Coleman-adapted
Kolyvagin systems that we develop here, we deduce parts of Perrin-Riou's main
conjecture from an explicit reciprocity conjecture.
