Related papers: Iterative q difference Galois Theory

Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable…

Differential equations have arithmetic analogues in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations and the present paper is concerned with the "linear" ones. The equations…

We prove in this paper an uniform surjectivity result for Galois representations associated with non-CM $\mathbb{Q}$-curves over imaginary quadratic fields, using various tools for the proof, such as Mazur's method, isogeny theorems,…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be a number field of degree four that is Galois over $\mathbb{Q}$. The goal of this article is to classify the different isomorphism types of $E(K)_{\text{tors}}$.

We describe an explicit construction of galoisian Stokes operators for irregular linear q-difference equations. These operators are parameterized by the points of an elliptic curve minus a finite set of singularities. Taking residues at…

This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generaliza- tion of linear systems, and the more general class of differential equations adimitting…

An exterior derivative, inner derivation, and Lie derivative are introduced on the quantum group $GL_{q}(N)$. $SL_{q}(N)$ is then found by constructing matrices with determinant unity, and the induced calculus is found.

The equivariant `main conjecture' of Iwasawa theory is shown to hold for a Galois extension $K/k$ of number fields with Galois group an $l$-adic pro-$l$ Lie group of dimension 1 containing an abelian subgroup of index $l$, provided that…

In this paper we compute the difference Galois groups of the Lie-irreducible regular singular generalized q-hypergeometric equations of order 3 with q-real parameters by using a density theorem due to Sauloy. In contrast with the…

We give model theoretic accounts and proofs of the existence and uniqueness of differential Galois extensions with no new constants, for logarithmic differential equations over a differential field K, when the field C of constants of K is…

A Galois correspondence theorem is proved for the case of inverse semigroups acting orthogonally on commutative rings as a consequence of the Galois correspondence theorem for groupoid actions. To this end, we use a classic result of…

Gauge theory on the q-deformed two-dimensional Euclidean plane R^2_q is studied using two different approaches. We first formulate the theory using the natural algebraic structures on R^2_q, such as a covariant differential calculus, a…

We introduce and study a class of field extensions that we call pre-Galois; viz. extensions that become Galois after some linearly disjoint Galois base change. Among them are geometrically Galois extensions of k(T), with k a field:…

In his earlier work, the author introduced a group theory question that arises in the study of iterated Galois groups of post-critically finite quadratic polynomials. In this paper, we prove the first non-trivial results on this question.

The aim of this article is to provide a method to prove the irreducibility of non-linear ordinary differential equations by means of the differential Galois group of their variational equations along algebraic solutions. We show that if the…

Motives and automorphic forms of arithmetic type give rise to Galois representations that occur in {\it compatible families}. These compatible families are of p-adic representations with p varying. By reducing such a family mod p one…

Given a non-negative integer $q$, we study two different notions of the $q$-capability of Lie algebras via the non-abelian $q$-exterior product of Lie algebras. The first is related to the $q$-crossed modules and inner $q$-derivations, and…

Studied is a generalization of Zagier's q-series identity. We introduce a generating function of L-functions at non-positive integers, which is regarded as a half-differential of the Andrews--Gordon q-series. When q is a root of unity, the…

For finite Galois extension fields defined by odd degree irreducible polynomials over algebraic integer ring, we observe "Reciprocity Law" through Jacobian Variety by embedding all roots of the polynomials into 2-torsion points of Jacobian…

Using Galois theory of functional equations, we give a new proof of the main result of the paper "Transcendental transcendency of certain functions of Poincar\'e" by J.F. Ritt, on the differential transcendence of the solutions of the…