Integral Iwasawa theory of Galois representations for non-ordinary primes

Abstract

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.