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We prove the finiteness and compatibility with base change of the (φ,Γ)(\\varphi , \\Gamma )-cohomology and the Iwasawa cohomology of arithmetic families of (φ,Γ)(\\varphi , \\Gamma )-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.

Continuing the study of the Iwasawa theory of symmetric powers of CM modular forms at supersingular primes begun by the first author and Antonio Lei, we prove a Main Conjecture equating the “admissible”p-adicL-functions to the characteristic ideals of “finite-slope” Selmer modules constructed by the second author. As a key ingredient, we improve Rubin’s result on the Main Conjecture of Iwasawa theory for imaginary quadratic fields to an equality at inert primes.

An abstract formulation of Harder-Narasimhan theory is stated without proof by L. Fargues, and I found it helpful to write it all out.

Given a rank two trianguline family of \\((\\varphi,\\Gamma)\\)-modules having a noncrystalline semistable member, we compute the Fontaine--Mazur \\(\\mathcal{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 Qp-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 (phi, Gamma)-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.

We present two results about Selmer groups. Given a torsion p-adic Galois representation A of a number field K, the Selmer group of A over K is the subspace of Galois cohomology H1( GK, A) consisting of cycles c satisfying certain local conditions, i.e. such that the restrictions res v(c) ∈ H1( Gv, A) to decomposition groups Gv (for places v of K) lie in distinguished subspaces Lv ⊆ H 1(Gv, A). These groups are conjecturally related to algebraic cycles (à la Shafarevich–Tate) on the one hand, and on the other to special values of L-functions (à la Bloch–Kato). Our first result shows how, using a global symmetry (the sign of functional equation under Tate global duality), one can produce increasingly large Selmer groups over the finite subextensions of a [special characters omitted]-extension of K. Our second result gives a new characterization of the Selmer group, namely of the local condition Lv for v | p. It uses (ϕ, [special characters omitted])-modules over Berger's Robba ring [special characters omitted] to give a vast generalization of the well-known \"ordinary\" condition of Greenberg to the nonordinary setting. We deduce a definition of Selmer groups for overconvergent modular forms (of finite slope). We also propose a program, using variational techniques, that would give a definition of the Selmer group along the eigencurve of Coleman–Mazur, including notably its nonordinary locus.

We generalize to the finite-slope setting several techniques due to Nekovar concerning the parity conjecture for self-dual motives. In particular we show that, for a \\(p\\)-adic analytic family, with irreducible base, of symplectic self-dual global Galois representations whose \\((\\varphi,\\Gamma)\\)-modules at places lying over \\(p\\) satisfy a Panchishkin condition, the validity of the parity conjecture is constant among all specializations that are pure. As an application, we extend some other results of Nekovar for Hilbert modular forms from the ordinary case to the finite-slope case.

Continuing the study of the Iwasawa theory of symmetric powers of CM modular forms at supersingular primes begun by the first author and Antonio Lei, we prove a Main Conjecture equating the \"admissible\" \\(p\\)-adic \\(L\\)-functions to characteristic ideals of \"finite-slope\" Selmer modules constructed by the second author. As a key ingredient, we improve Rubin's result on the Main Conjecture of Iwasawa theory for imaginary quadratic fields to an equality at inert primes.

Let \\(p\\) be a prime number, and let \\(K\\) be a \\(p\\)-adic local field. We study a class of semistable \\(p\\)-adic Galois representations of \\(K\\), which we call {\\it triangulordinary} because it includes the ordinary ones yet allows non-étale behavior in the associated \\((\\phi,\\Gamma_K)\\)-modules over the Robba ring. Our main result provides a description of the Bloch--Kato local condition of such representations. We also propose a program, using variational techniques, that would give a definition of the Selmer group along the eigencurve of Coleman--Mazur, including notably its nonordinary locus.

Let \\(p\\) be an odd prime, and let \\(K/K_0\\) be a quadratic extension of number fields. Denote by \\(K_\\pm\\) the maximal \\(\\mathbb{Z}_p\\)-power extensions of \\(K\\) that are Galois over \\(K_0\\), with \\(K_+\\) abelian over \\(K_0\\) and \\(K_-\\) dihedral over \\(K_0\\). In this paper we show that for a Galois representation over \\(K_0\\) satisfying certain hypotheses, if it has odd Selmer rank over \\(K\\) then for one of \\(K_\\pm\\) its Selmer rank over \\(L\\) is bounded below by \\([L:K]\\) for \\(L\\) ranging over the finite subextensions of \\(K\\) in \\(K_\\pm\\). Our method or proof generalizes a method of Mazur--Rubin, building upon results of Nekovář, and applies to abelian varieties of arbitrary dimension, (self-dual twists of) modular forms of even weight, and (twisted) Hida families.