We prove the finiteness and compatibility with base change of the (ϕ, Γ)-cohomology and the Iwasawa cohomology of arithmetic families of (ϕ, Γ)-modules. Using this finiteness theorem, we show that a family of Galois representations that is densely pointwise refined in the sense of Mazur is actually trianguline as a family over a large subspace. In the case of the Coleman-Mazur eigencurve, we determine the behavior at all points.

We develop a theory of Selmer groups for analytic families of Galois representations, which are only assumed "ordinary" on the level of their underlying (ϕ,)modules. Our approach brings the finite-slope nonordinary case of Iwasawa theory onto an equal footing with ordinary cases in which p is inverted.

Given a rank two trianguline family of (ϕ, Γ)-modules having a noncrystalline semistable member, we compute the Fontaine-Mazur L-invariant of that member in terms of the logarithmic derivative, with respect to the Sen weight, of the value at p of the trianguline parameter. This generalizes prior work, in the case of Galois representations, due to Greenberg-Stevens and Colmez.

Let K be a complete discretely valued field of mixed characteristics (0, p) with perfect residue field. One of the central objects of study in p-adic Hodge theory is the category of continuous representations of the absolute Galois group of K on finite-dimensional Q p -vector spaces. In recent years, it has become clear that this category can be studied more effectively by embedding it into the larger category of (ϕ, Γ)-modules; this larger category plays a role analogous to that played by the category of vector bundles on a compact Riemann surface in the Narasimhan-Seshadri theorem on unitary representations of the fundamental group of said surface. This category turns out to have a number of distinct natural descriptions, which on one hand suggests the naturality of the construction, but on the other hand forces one to use different descriptions for different applications. We provide several of these descriptions and indicate how to translate certain key constructions, which were originally given in the context of modules over power series rings, to the more modern context of perfectoid algebras and spaces.

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