Duality theory of p-adic Hopf algebras

We show the monoidal functoriality of Schikhof duality, and cultivate new duality theory of p-adic Hopf algebras. Through the duality, we introduce two sorts of p-adic Pontryagin dualities. One is a duality between discrete Abelian groups and affine formal group schemes of specific type, and the other one is a duality between profinite Abelian groups and analytic groups of specific type. We extend Amice transform to a p-adic Fourier transform compatible with the second p-adic Pontryagin duality. As applications, we give explicit presentations of a universal family of irreducible p-adic unitary Banach representations of the open unit disc of the general linear group and its q-deformation in the case of dimension ۲.