Related papers: On sigma-delta-Picard-Vessiot extensions

We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automorphism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness,…

We present a characterization of differential Picard-Vessiot extensions which is analogous to the field theoretic characterization of Galois extensions in classical Galois theory.

These are the notes for an undergraduate course at the University of Edinburgh, 2021-2023. Assuming basic knowledge of ring theory, group theory and linear algebra, the notes lay out the theory of field extensions and their Galois groups,…

We construct small parabolic eigenvarieties for holomorphic Siegel cuspforms of genus $2$ and study families of Galois representations attached to them in the spirit of Bella\"iche--Chenevier. In the course, we introduce the notion of…

In previous papers, we defined $q$-analogues of alien derivations for linear analytic $q$-difference equations with integral slopes and proved a density theorem (in the Galois group) and a freeness theorem. In this paper, we completely…

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…

We classify all cubic function fields over any finite field, particularly developing a complete Galois theory which includes those cases when the constant field is missing certain roots of unity. In doing so, we find criteria which allow…

Let $E/F$ be a cyclic Galois extension of degree $p^l$ with Galois group $G$. It is shown that the Galois module structure of both sides of the Kummer pairing (for Kummer extensions of $E$) are the same. In other words, we show that the…

We investigate the Liouvillian integrability of Hamiltonian systems describing a universe filled with a scalar field (possibly complex). The tool used is the differential Galois group approach, as introduced by Morales-Ruiz and Ramis. The…

We study the arithmetic of curves and Jacobians endowed with the action of a finite group $G$. This includes a study of the basic properties, as $G$-modules, of their $\ell$-adic representations, Selmer groups, rational points and…

We introduce the notion of differential torsors, which allows the adaptation of constructions from algebraic geometry to differential Galois theory. Using these differential torsors, we set up a general framework for applying patching…

Given a number field $F$, a finite group $G$ and an indeterminate $T$, {\it{a $G$-parametric extension over $F$}} is a finite Galois extension $E/F(T)$ with Galois group $G$ and $E/F$ regular that has all the Galois extensions of $F$ with…

For a differential operator $L$ of order $n$ over $C(z)$ with a finite (differential) Galois group $G\subset {\rm GL}(C^n)$, there is an algorithm, by M. van Hoeij and J.-A.~Weil, which computes the associated evaluation of the invariants…

We provide new proofs of two key results of p-adic Hodge theory: the Fontaine-Wintenberger isomorphism between Galois groups in characteristic 0 and characteristic p, and the Cherbonnier-Colmez theorem on decompletion of (phi,…

We define vector fields, leaves and trajectories for schemes. With these tools, we are able to give a geometrical interpretation and to generalize several results of differential Galois theory and constructions on differential schemes. We…

We investigate the first two Galois cohomology groups of $p$-extensions over a base field which does not necessarily contain a primitive $p$th root of unity. We use twisted coefficients in a systematic way. We describe field extensions…

We construct extensions of the field of rational numbers with the Galois group G_2(F_p) by reducing p-adic representations attached to automorphic representations.

Let $ K$ be a field admitting a Galois extension $L$ of degree $n$, denoting the Galois group as $G = \gal(L/K)$. Our focus lies on the space $\sym_K(L)$ of symmetric $K$-bilinear forms on $L$. We establish a decomposition of $\sym_K(L)$…

We introduce Galois Theory for Hopf-Galois Extensions proving existence of a Galois connection between subalgebras of an H-comodule algebra and generalised quotients of the Hopf algebra H. Moreover, we show that these quotients Q which…

In this paper, we give a geometrization and a generalization of a lemma of differential Galois theory. This geometrization, in addition of giving a nice insight on this result, offers us the occasion to investigate several points of…