Related papers: Split embedding problems over the open arithmetic …

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is…

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 that any irreducible representation $\rho$ of a finite group $G$ of exponent $n$, realisable over $\mathbb{R}$, is realisable over the field $E:=\mathbb{Q}(\zeta_n)\cap\mathbb{R}$ of real cyclotomic numbers of order $n$, and…

Let E be an elliptic curve over the rationals without complex multiplication. The absolute Galois group of Q acts on the group of torsion points of E, and this action can be expressed in terms of a Galois representation rho_E:Gal(Qbar/Q)…

The aim of the inverse Galois problem is to find extensions of a given field whose Galois group is isomorphic to a given group. In this article, we are interested in subgroups of GL(2,Z/nZ) where n is an integer. We know that, in general,…

Let $\Ga$ be a connected, solvable linear algebraic group over a number field~$K$, let $S$ be a finite set of places of~$K$ that contains all the infinite places, and let $\theints$ be the ring of $S$-integers of~$K$. We define a certain…

Let $G$ be a finite group. Then there exists a first-order statement $S(G)$ in the language of rings without parameters and depending only on $G$ such that, for any field $K$, we have that $K\models S(G)$ if and only if $K$ has a Galois…

For a finite group $G$, we introduce a generalization of norm relations in the group algebra $\mathbb Q[G]$. We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the…

In this paper, we prove new instances of the inverse Galois problem over global function fields for finite groups of Lie type. This is done by constructing compatible systems of $\ell$-adic Galois representations valued in a semisimple…

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…

Let $K$ be a number field and $K_{ur}$ be the maximal extension of $K$ that is unramified at all places. In a previous article, the first author found three real quadratic fields $K$ such that $Gal(K_{ur}/K)$ is finite and nonabelian simple…

We introduce a category of dual pairs of finite locally free algebras over a ring. This gives an efficient way to represent finite locally free commutative group schemes. We give a number of algorithms to compute with dual pairs of…

We give an explicit construction of global Galois gerbes constructed more abstractly by Kaletha to define global rigid inner forms. This notion is crucial to formulate Arthur's multiplicity formula for inner forms of quasi-split reductive…

We prove very general index formulae for integral Galois modules, specifically for units in rings of integers of number fields, for higher K-groups of rings of integers, and for Mordell-Weil groups of elliptic curves over number fields.…

We construct \'etale generalized Heisenberg group covers of hyperelliptic curves over number fields. We use these to produce infinite families of quadratic extensions of cyclotomic fields that admit everywhere unramified generalized…

We provide an infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $SL_2(7)$. To my knowledge, this is the first instance of infinitely many such realizations for a perfect group which is…

In this paper, we introduce and develop the circle embedding method. This method hinges essentially on a combinatorial-geometric structure which we choose to call circles of partition. We provide applications in the context of problems that…

This paper develops a harmonic Galois theory for finite graphs, thereby classifying harmonic branched $G$-covers of a fixed base $X$ in terms of homomorphisms from a suitable fundamental group of $X$ together with $G$-inertia structures on…

Given a finite transitive permutation group $G$, we investigate number fields $F/\mathbb{Q}$ of Galois group $G$ whose discriminant is only divisible by small prime powers. This generalizes previous investigations of number fields with…

Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division…