Difference Galois Theory For The "Applied" Mathematician

Galois Groups of Difference Equations of Order Two on Elliptic Curves

This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from transcendence theory due to Schneider allow us to identify a large class of discrete Lamé equations with difference Galois group $\operatorname{GL}_{2}(\mathbb C)$.

A density theorem for the difference Galois groups of regular singular Mahler equations

The difference Galois theory of Mahler equations is an active research area. The present paper aims at developing the analytic aspects of this theory. We first attach a pair of connection matrices to any regular singular Mahler equation. We then show that these connection matrices can be used to produce a Zariski-dense subgroup of the difference Galois group of any regular singular Mahler equation.

Intrinsic approach to Galois theory of q-difference equations, with the preface to Part 4 "The Galois D-groupoid of a q-difference system'' by Anne Granier

We give a complete answer to the analogue of Grothendieck conjecture on p-curvatures for q-difference equations defined over K(x), where K is any finitely generated extension of Q and q\in K can be either a transcendental or an algebraic number. This generalizes the results in [DV02], proved under the assumption that K is a number field and q an algebraic number. The results also hold for a field K which is a finite extension of a purely transcendental extension k(q) of a perfect field k. In Part 3, we consider two Galois groups attached to a q-difference module M over K(x): (1) the intrinsic …

Descent for differential Galois theory of difference equations. Confluence and q-dependency

The present paper essentially contains two results that generalize and improve some of the constructions of [arXiv:0801.1493]. First of all, in the case of one derivation, we prove that the parameterized Galois theory for difference equations constructed in [arXiv:0801.1493] can be descended from a differentially closed to an algebraically closed field. In the second part of the paper, we show that the theory can be applied to deformations of q-series, to study the differential dependency with respect to x\frac{d}{dx} and q\frac{d}{dq}. We show that the parameterized difference Galois group (w…

Serre's Modularity Conjecture

These are the lecture notes from a five-hour mini-course given at the Winter School on Galois Theory held at the University of Luxembourg in February 2012. Their aim is to give an overview of Serre's modularity conjecture and of its proof by Khare, Wintenberger, and Kisin, as well as of the results of other mathematicians that played an important role in the proof. Along the way we remark on some recent (as of 2012) work concerning generalizations of the conjecture.

On the Grothendieck conjecture on p-curvatures for q-difference equations

In the present paper, we give a q-analogue of the Grothendieck conjecture on p-curvatures for q-difference equations defined over the field of rational function K(x), where K is a finite extension of a field of rational functions k(q), with k perfect. Then we consider the generic (also called intrinsic) Galois group in the sense of N. Katz. The result in the first part of the paper lead to a description of the generic Galois group through the properties of the functional equations obtained specializing q on roots of unity. Although no general Galois correspondence holds in this setting, in the…

A general theory of André's solution algebras

We extend Yves André's theory of solution algebras in differential Galois theory to a general Tannakian context. As applications, we establish analogues of his correspondence between solution fields and observable subgroups of the Galois group for iterated differential equations in positive characteristic and for difference equations. The use of solution algebras in the difference algebraic context also allows a new approach to recent results of Philippon and Adamczewski--Faverjon in transcendence theory.

关于完全幂与素数幂的差

On differences of perfect powers and prime powers

本研究提出一种结合Frey-Hellegouarch曲线伽罗瓦表示、Thue-Mahler方程有效求解及改进对数下界的方法，显式确定完全幂与素数幂的差的无平方因子部分对应元组，并求解了指定参数范围的情形。

关于zeta函数的Ser表示与Hasse表示的三则注记

Three notes on Ser's and Hasse's representations for the zeta-functions

本文研究zeta函数的Ser与Hasse级数表示，证明二者等价，推导了涉及多种特殊数与多项式的zeta函数、Hurwitz zeta函数及Dirichlet级数的新级数表示，还得到Stieltjes常数等的新展开式。