Related papers: A group-theoretic generalization of the $p$-adic l…

Let $p$ be a prime, and let $K$ be a finite extension of $\mathbf{Q}_p$, with absolute Galois group $\cal{G}_K$. Let $\pi$ be a uniformizer of $K$ and let $K_\infty$ be the Kummer extension obtained by adjoining to $K$ a system of…

Here we show that, given a finite homological system $({\cal P},\leq,\{\Delta_u\}_{u\in {\cal P}})$ for a finite-dimensional algebra $\Lambda$ over an algebraically closed field, the category ${\cal F}(\Delta)$ of $\Delta$-filtered modules…

We generalize the concept of rigid inner forms, defined by Kaletha in [Kal16], to the setting of a local function field $F$ in order state the local Langlands conjectures for arbitrary connected reductive groups over $F$. To do this, we…

We introduce the notion of the $\infty$-category of (complete) derived $G$-graded modules over a $G$-graded ring $R$ for a torsion-free abelian group $G$, and we study its foundational properties. Moreover, we prove a categorical…

If in a given rank $r$, there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number $\ell$, there is an absolutely…

Let F be the function field of a curve over a complete discretely valued field K. Let G be a semisimple simply connected linear algebraic group over F of type An. We give a description of the obstruction to local global principle for…

To each category C of modules of finite length over a complex simple Lie algebra g, closed under tensoring with finite dimensional modules, we associate and study a category Aff(C)_\kappa of smooth modules (in the sense of Kazhdan and…

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 $G$ be a connected reductive group over a non-archimedean local field $F$ of residue characteristic $p$, $P$ be a parabolic subgroup of $G$, and $R$ be a commutative ring. When $R$ is artinian, $p$ is nilpotent in $R$, and…

We consider the rational map $F$ defined by the quotient of products of lines in general position and we study the monodromy problem and tangential center-focus problem for the fibration associated with $F$. Thus, we study the submodule of…

We begin to study Steinberg groups associated with a locally isotropic reductive group $G$ over a arbitrary ring. We propose a construction of such a Steinberg group functor as a group object in a certain completion of the category of…

We use Scholze's framework of diamonds to gain new insights in correspondences between $p$-adic vector bundles and local systems. Such correspondences arise in the context of $p$-adic Simpson theory in the case of vanishing Higgs fields. In…

For a smooth group scheme $G$ over an extension of $\mathbf{Z}_p$ such that the generic fiber of $G$ is reductive, we study the generic fiber of the Galois deformation ring for a $G$-valued mod $p$ representation of the absolute Galois…

We study the $p$-adic analogue of the $\ell$-adic hypergeometric sheaves for reductive groups, called the hypergeometric $\mathscr{D}^{\dagger}(\infty)$-modules. They are overholonomic objects in the derived category of arithmetic…

We give a construction of "integral local Shimura varieties" which are formal schemes that generalize the well-known integral models of the Drinfeld $p$-adic upper half spaces. The construction applies to all classical groups, at least for…

We prove a conjecture of J. Carlson, N. Mazza and J. Th\'evenaz; namely, we will prove that if $G$ is a finite $p$-nilpotent group which contains a non-cyclic elementary Abelian $p$-subgroup and $k$ is an algebraically closed field of…

Let $F$ be a function field over $\mathbb{F}_q$, $A$ its ring of regular functions outside a place $\infty$ and $\mathfrak{p}$ a prime ideal of $A$. First, we develop Hida theory for Drinfeld modular forms of rank $r$ which are of slope…

Let K be a p-adic field and F the function field of a curve over K. Let G be a connected linear algebraic group over F of classical type. Suppose the prime p is a good prime for G. Then we prove that projective homogeneous spaces under G…

Let F be a locally compact non-archimedean field and G the group of F-rational points of an algebraic group assumed to be defined over F, semisimple, simply connected and of F-rank 1. Let pi be a complex irreducible supercuspidal…

We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a k_\omega-space, or locally k_\omega. As a first application,…