Related papers: Local Monodromy of 1-Dimensional p-Divisible Group…

In this work we compute the universal framed deformation functor for a reducible Galois representation $\rho$ given by direct sum of 2-dimensional representations $\rho_i$ coming from p-divisible groups. We impose the local conditions of…

Let $G$ be a finite classical group of Lie type of rank $\ell$, defined over a field of characteristic $p>2$. In this work, we classify the irreducible representations of $G$ whose dimensions are bounded by a constant proportional to…

Let $n \geq 2$ and $p$ be a prime. Let $K$ be a number field and consider two Galois representations $\rho_1, \rho_2 : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_n(\mathbb{Z}_p)$ having residual image a $p$-group. We explain…

This Ph.D. thesis belongs to the realm of mod $p$ representation theory of $p$-adic groups. The main object of study is the inner form of the general linear group $\mathrm{GL}(m,D)$ where $D$ is a division algebra over a non-Archimedean…

We consider the generalized character $\Psi_{1,p,G}$ of a finite group $G$ which vanishes on all $p$-singular elements of $G$ and whose value at each $p$-regular $y \in G$ is the number of $p$-elements of $C_{G}(y)$. We conjecture that this…

Denote by $\mathfrak{o}$ the valuation ring of a non-Archimedean local field with prime ideal $\mathfrak{p}$ and finite residue field, and let $r\geq 1$ be an integer. We prove that for every smooth affine group scheme $G$ over…

Let $p$ be a prime number, $n>2$ an integer, and $F$ a CM field in which $p$ splits completely. Assume that a continuous automorphic Galois representation…

In this paper, we study the Galois representations attached to products of Drinfeld modules. As an analogue of Serre's classical result on the images of Galois representations associated with products of elliptic curves, we prove that for…

We study deformation theory of mod $p$ Galois representations of $p$-adic fields with values in generalised reductive group schemes, such as $L$-groups and $C$-groups. We show that the corresponding deformation rings are complete…

An algebraic extension of the rational numbers is said to have the $\textit{Bogomolov property}$ (B) if the absolute logarithmic Weil height of its non-torsion elements is uniformly bounded from below. Given a continuous representation…

We show that if $\{\rho_{\ell}\}$ is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation $\overline{\rho}_{\ell}$ is absolutely irreducible for $\ell$ in a density 1 set of…

In this paper, we classify all continuous Galois representations $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{Q}}_p)$ which are unramified outside $\{p,\infty\}$ and locally induced at $p$, under…

Let $K$ be a totally real field and $\pi$ be a regular algebraic polarized cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$. Let $\{\rho_{\pi,\lambda}:\mathrm{Gal}_K\to\mathrm{GL}_n(\overline E_\lambda)\}_\lambda$ be the…

We prove the compatibility of local and global Langlands correspondences for $GL_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne and Scholze. More precisely, let $r_p(\pi)$ denote an…

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because…

Suppose $\rho_1, \rho_2$ are two $\ell$-adic Galois representations of the absolute Galois group of a number field, such that the algebraic monodromy group of one of the representations is connected and the representations are locally…

Let $\mathbb{F}_q$ be the finite field with $q$ elements, $F:=\mathbb{F}_q(T)$ and $F^{\operatorname{sep}}$ a separable closure of $F$. Set $A$ to denote the polynomial ring $\mathbb{F}_q[T]$. Let $\mathfrak{p}$ be a non-zero prime ideal of…

Let $\rho_p$ be an $n$-dimensional non-critical semistable $p$-adic Galois representation of the absolute Galois group of $\mathrm{Q}_p$ with regular Hodge--Tate weights. Let $\mathrm{D}$ be the associated $(\varphi,\Gamma)$-module over the…

We take some initial steps towards illuminating the (hypothetical) $p$-adic local Langlands functoriality principle relating Galois representations of a $p$-adic field $L$ and admissible unitary Banach space representations of $G(L)$ when…

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $\rho$ be an $n$-dimensional (non-critical generic) crystabelline representation of the absolute Galois group of $K$ of regular Hodge-Tate weights. We associate to $\rho$ an explicit…