Related papers: Families of two dimensional modular $(\varphi,\Gam…

Let $p$ be an odd prime, and $\mathbf{Q}_{p^f}$ the unramified extension of $\mathbf{Q}_p$ of degree $f$. In this paper, we reduce the problem of constructing strongly divisible modules for $2$-dimensional semi-stable non-crystalline…

We construct moduli stacks of two-dimensional mod p representations of the absolute Galois group of a p-adic local field, and relate their geometry to the weight part of Serre's conjecture for GL(2).

Let $K$ be a finite extension of $\mathbf{Q}_p$ and let $G_K = \mathrm{Gal}(\bar{\mathbf{Q}}_p/K)$. There is a very useful classification of $p$-adic representations of $G_K$ in terms of cyclotomic $(\varphi,\Gamma)$-modules (cyclotomic…

We propose a p-adic Langlands correspondence in families.

We consider the moduli space, in the sense of Kisin, of finite flat models of a 2-dimensional representation with values in a finite field of the absolute Galois group of a totally ramified extension of $mathbb{Q}_p$. We determine the…

The submodule structure of mod $p$ principal series representations of $\mathrm{GL}_2(k)$, for $k$ a finite field of characteristic $p$, was described by Bardoe and Sin and has played an important role in subsequent work on the mod $p$…

The construction of the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbf{Q}_p)$ uses in an essential way Fontaine's theory of cyclotomic $(\varphi,\Gamma)$-modules. Here \emph{cyclotomic} means that $\Gamma =…

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. We exhibit slices of the representation theory of $\Lambda$ that are always classifiable in stringent geometric terms. Namely, we prove that, for any…

We investigate the relation between p-adic Galois representations and overconvergent (phi,Gamma)-modules in families. Especially we construct a natural open subspace of a family of (phi,Gamma)-modules, over which it is induced by a family…

Let L be a non-archimedean local field of characteristic 0. We present a variant of the theory of (\phi,\Gamma)-modules associated with Lubin-Tate groups, developed by Kisin and Ren [Ki-Re], in which we replace the Lubin-Tate tower by the…

We study the cohomology of families of $(\varphi,\Gamma)$-modules with coefficients in pseudoaffinoid algebras. We prove that they have finite cohomology, and we deduce an Euler characteristic formula and Tate local duality. We classify…

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…

We construct noncommutative multidimensional versions of overconvergent power series rings and Robba rings. We show that the category of \'etale $(\varphi,\Gamma)$-modules over certain completions of these rings are equivalent to the…

Let $K$ be any finite extension of $Q_{p}$, $L$ any finite Galois extension of $K$ and $E$ any finite large enough coefficient field containing $L$. We classify two-dimensional, F-semistable $E$-representations of $G_{K}$, by listing the…

Let $F/{\mathbb Q}_p$ be a finite field extension, let $k$ be a finite field extension of the residue field of $F$. Generalizing the $\psi$-lattices which Colmez constructed in \'{e}tale $(\varphi,\Gamma)$-modules over $k[[t]][t^{-1}]$, we…

Berger and Colmez introduced a theory of families of overconvergent \'etale (Phi,Gamma)-modules associated to families of p-adic Galois representations over p-adic Banach algebras. However, in contrast with the classical theory of…

We give a classification of rank one $(\varphi,\Gamma)$-modules with coefficients in a $p$-adically complete $\mathbf{Z}_p$-algebra. As a consequence, we obtain a new proof of Proposition 7.2.17 in {arXiv:1908.07185}, which gives an…

In this paper, we give a complete classification of extensions of finite irreducible conformal modules over rank two Lie conformal algebras.

We develop a new method for analyzing moduli problems related to the stack of pure coherent sheaves on a polarized family of projective schemes. It is an infinite-dimensional analogue of geometric invariant theory. We apply this to two…

We show that an infinite family of odd complex 2-dimensional Galois representations ramified at 5 having nonsolvable projective image are modular, thereby verifying Artin's conjecture for a new case of examples. Such a family contains the…