Related papers: Raccord sur les espaces de Berkovich

We prove, using a theorem of Northcott, that if a number field K with s real embeddings and 2t complex ones has a group of units U such that all elements in U have all its complex conjugates of same absolute value, then one necessarily has…

We consider certain strengthenings of property (T) relative to Banach spaces that are satisfied by high rank Lie groups. Let X be a Banach space for which, for all k, the Banach--Mazur distance to a Hilbert space of all k-dimensional…

We introduce the class of strongly sofic monoids. This class of monoids strictly contains the class of sofic groups and is a proper subclass of the class of sofic monoids. We define and investigate sofic topological entropy for actions of…

We compute the Galois group of the splitting field $F$ of any irreducible and separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with characteristic different from two. The proofs require to distinguish between two cases:…

In this manuscript, we apply patching methods to give a positive answer to the inverse differential Galois problem over function fields over Laurent series fields of characteristic zero. More precisely, we show that any linear algebraic…

If A is a finite dimensional nilpotent associative algebra over a finite field k, the set G=1+A of all formal expressions of the form 1+a, where a is an element of A, has a natural group structure, given by (1+a)(1+b)=1+(a+b+ab). A finite…

If k is a commutative field and G a reductive (connected) algebraic group over k, we give bounds for the orders of the finite subgroups of G(k); these bounds depends on the type of G and on the Galois groups of the cyclotomic extensions of…

We prove that every finite subgroup of $GL_{2}(\mathbb{R})$ can be realized as the Veech group of some translation surface.

Given a smooth geometrically connected curve $C$ over a field $k$ and a smooth commutative group scheme $G$ of finite type over the function field $K$ of $C$ we study the Tate--Shafarevich groups given by elements of $H^1(K,G)$ locally…

A central conjecture in inverse Galois theory, proposed by D\`{e}bes and Deschamps, asserts that every finite split embedding problem over an arbitrary field can be regularly solved. We give an unconditional proof of a consequence of this…

Motivated by the work of Liu, we study certain canonical quotients of $G_{\emptyset}^T(K)$ -- the Galois group of the maximal unramified extension of a global field $K$ that is split completely at a finite nonempty set of places in $T$ --…

Let $K$ be the function field of a smooth projective geometrically integral curve over a finite extension of $\mathbb{Q}_p$. Following the works of Harari, Scheiderer, Szamuely, Izquierdo, and Tian, we study the local-global and weak…

Let $k/\mathbb F_p$ denote a finite field. For any split connected reductive group $G/W(k)$ and certain CM number fields $F$, we deform certain Galois representations $\overline\rho:Gal(\overline F/F) \to G(k)$ to continuous families…

Let $K$ be a field finitely generated over the field of rational numbers, $K(c)$ the extension of $K$ obtained by adjoining all roots of unity, $L$ an infinite Galois extension of $K$, $X$ an abelian variety defined over $K$. We prove that…

We show that every effectively closed action of a finitely generated group $G$ on a closed subset of $\{0,1\}^{\mathbb{N}}$ can be obtained as a topological factor of the $G$-subaction of a $(G \times H_1 \times H_2)$-subshift of finite…

Let $f(x)$ be a monic polynomial in $\dZ[x]$ with no rational roots but with roots in $\dQ_p$ for all $p$, or equivalently, with roots mod $n$ for all $n$. It is known that $f(x)$ cannot be irreducible but can be a product of two or more…

This paper gives a quick overview of the author's recent result that all finitely presented groups are QSF.

Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types…

For a quadratic field K, we investigate continuous mod p representations of the absolute Galois groups of K that are unramified away from p and infinity. We prove that for certain pairs (K,p), there are no such irreducible representations.…

We prove the existence of a Galois closure for towers of torsors under finite group schemes over a proper, geometrically connected and geometrically reduced algebraic stack $X$ over a field $k$. This is done by describing the Nori…