Related papers: Chasing maximal pro-p Galois groups via 1-cyclotom…

We exhibit the first examples of residually finite non-linear Gromov hyperbolic groups. Our examples are constructed as amalgamated products of torsion-free cocompact lattices in the rank 1 Lie group $\mathrm{Sp}(d,1)$, $d\geq 2$ along…

Using Galois representations attached to elliptic curves, we construct Galois extensions of $\mathbb{Q}$ with group $GL_2(p)$ in which all decomposition groups are cyclic. This is the first such realization for all primes $p$.

We propose an approach to study non-Abelian Iwasawa theory, using the idea of Johnson homomorphisms in low dimensional topology. We introduce arithmetic analogues of Johnson homomorphisms/maps, called the p-Johnson homomorphisms/maps,…

Let p be a prime. We classify finitely generated pro-p groups G which satisfy d(H) = d(G) for all open subgroups H of G. Here d(H) denotes the minimal number of topological generators for the subgroup H. Within the category of p-adic…

The structure of the Galois group of the maximal unramified p-extension of an imaginary quadratic field is restricted in various ways. In this paper we construct a family of finite 3-groups satisfying these restrictions. We prove several…

Let $f$ be an irreducible polynomial of prime degree $p\geq 5$ over $\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens H\"{o}chsmann (1999), we show that if $p\geq 4k+1$ then $\Gal(f/\QQ)$ is isomorphic to $A_{p}$ or…

This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur…

We present here a version of the Artin-Schreier polynomial that works in any characteristic. Let $C_p$ be the cyclic group of prime order $p$. Equivalently, we prove one can lift degree $p$ cyclic $C_p$ extensions over local rings $R,M$…

Let $p$ be a prime, $D$ a finite dimensional noncommutative division $\mathbb{Q}_p$-algebra, and $SL_1(D)$ the group of elements of $D$ of reduced norm $1$. When the center of $D$ is $\mathbb{Q}_p$, we prove that no open subgroup of…

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient. Motivated by the study of Hausdorff dimension, we show that finitely generated closed subgroups $H$ of infinite index in $G$ never contain any infinite subgroups $K$…

We introduce the first provably efficient algorithm to check if a finitely generated subgroup of an almost simple semi-simple group over the rationals is Zariski-dense. We reduce this question to one of computing Galois groups, and to this…

We compute the Galois groups of the reductions modulo the prime numbers $p$ of the generating series of Ap\'ery numbers, Domb numbers and Almkvist--Zudilin numbers. We observe in particular that their behavior is governed by congruence…

In this paper we compute the Galois cohomology of the pro-p completion of primitive link groups. Here, a primitive link group is the fundamental group of a tame link in the 3-sphere whose linking number diagram is irreducible modulo p (e.g.…

Given a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $\gamma$ of elements of $S$, one can associate an arboreal representation to $\gamma$, generalizing the case of iterating a single polynomial. We study…

Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$…

We show for all local fields $K/\mathbb{Q}_p$, with $p >3$, all representations $\bar\rho:G_K \to G_2(\bar{\mathbb{F}}_p)$ admit a crystalline lift $\rho: G_K\to G_2(\bar{\mathbb{Z}}_p)$, where $G_2$ is the exceptional Chevalley group of…

We determine the representation of the group of automorphisms for cyclotomic function fields in characteristic $p > 0$ induced by the natural action on the space of holomorphic differentials via construction of an explicit basis of…

Given an odd prime number $p$ and a Coxeter group $W$ such that the order of the product $st$ is prime to $p$ for every Coxeter generators $s,t$ of $W$, we prove that the $p$-local homology groups $H_k(W,\mathbb{Z}_{(p)})$ vanish for $1\leq…

For any odd prime $p$, the Galois group of the maximal unramified pro-$p$-extension of an imaginary quadratic field is a Schur $\sigma$-group. But Schur $\sigma$-groups can also be constructed and studied abstractly. We prove that if $p>3$,…

We establish automatic realizations of Galois groups among groups M\rtimes G, where G is a cyclic group of order p^n for a prime p and M is a quotient of the group ring Fp[G].