Related papers: The Realizability Problem with Inertia Conditions

We first prove the existence of minimally ramified p-adic lifts of 2-dimensional mod p representations, that are odd and irreducible, of the absolute Galois group of Q,in many cases. This is predicted by Serre's conjecture that such…

Let $n$ denote either a positive integer or $\infty$, let $\ell$ be a fixed prime and let $K$ be a field of characteristic different from $\ell$. In the presence of sufficiently many roots of unity in $K$, we show how to recover some of the…

This article is on the inverse Galois problem in Galois theory of linear iterative differential equations in positive characteristic. We show that it has an affirmative answer for reduced algebraic group schemes over any iterative…

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 extend Colmez's functor defined for $\operatorname{GL}_2(\mathbf{Q}_p)$ to the category of finitely generated smooth admissible mod-$p$ representations of the two-fold metaplectic cover of $\operatorname{GL}_2(\mathbf{Q}_p)$. We compute…

Let $X$ be a smooth projective connected curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Let $G$ be a finite group, $P$ a Sylow $p$-subgroup of $G$ and $N_G(P)$ its normalizer in $G$. We show…

A pro-$p$ group $G$ is called strongly Frattini-resistant if the function $H \mapsto \Phi(H)$, from the poset of all closed subgroups of $G$ into itself, is a poset embedding. Frattini-resistant pro-$p$ groups appear naturally in Galois…

For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K=Q, the answer is described by the long…

We prove that for any prime $p>2$, $q=p^\nu$ a power of $p$, $n\ge p$ and $G=S_n$ or $G=A_n$ (symmetric or alternating group) there exists a Galois extension $K/\mathbb F_q(T)$ ramified only over $\infty$ with $\mathrm{Gal}(K/\mathbb…

Let G be a finite group and V a finite-dimensional rational G-representation. We ask whether there exists a finite Galois extension L/K of number fields with Galois group G, an elliptic curve E/K, and a G-submodule of E(L) tensor Q…

The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that $\bar{\rho}:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow…

For various nonsolvable groups $G$, we prove the existence of extensions of the rationals $\mathbb{Q}$ with Galois group $G$ and inertia groups of order dividing $ge(G)$, where $ge(G)$ is the smallest exponent of a generating set for $G$.…

Let $G_{\mathbb{Q}_p}$ be the absolute Galois group of $\mathbb{Q}_p$ and let $L$ be a finite extension of $\mathbb{Q}_p$. Moreover let $\bar\rho:G_{\mathbb{Q}_p}\rightarrow GL_n(k_L)$ be a continous representation of $G_{\mathbb{Q}_p}$,…

Suppose that F/F+ is a CM extension of number fields in which the prime p splits completely and every other prime is unramified. Fix a place w|p of F. Suppose that rbar : Gal(F-bar/F) -> GL_3(Fp-bar) is a continuous irreducible Galois…

In 1960, L\'{a}szl\'{o} Fuchs posed the problem of determining which groups $G$ are realizable as the group of units in some ring $R$. In \cite{chebolu2022fuchs}, we investigated the following variant of Fuchs' problem, for abelian groups:…

This paper proves that if $E$ is a field, such that the Galois group $\mathcal{G}(E(p)/E)$ of the maximal $p$-extension $E(p)/E$ is a Demushkin group of finite rank $r(p)_{E} \ge 3$, for some prime number $p$, then $\mathcal{G}(E(p)/E)$…

Which groups can occur as the group of units in a ring? Such groups are called realizable. Though the realizable members of several classes of groups have been determined (e.g., cyclic, odd order, alternating, symmetric, finite simple,…

Let E be a CM number field, F its maximal totally real subfield, c the generator of Gal(E/F), p an odd prime totally split in E, and S a finite set of places of E containing the places above p. Let r : G_{E,S} --> GL_3(F_p^bar) be a…

In the present paper, we shall show that for any prime number p, every finite p-group occurs as the Galois Group of the maximal unramified p-extension over a certain number field of finite degree. We shall also show that for any given…

Given a reducible Galois representation $\overline{\rho}: G_{\mathbb{Q}} \rightarrow GL_2( \mathbb{F}_q)$ we show there exists an irreducible deformation $\rho : G_{\mathbb{Q}} \rightarrow GL_2 (\mathbb{W} [[T_1, T_2,.., T_r,....,]])$ of…