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# Embedding problem

About: Embedding problem is a research topic. Over the lifetime, 2873 publications have been published within this topic receiving 48403 citations.

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01 Jan 2003

TL;DR: In this paper, a large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, inverse problem and linear differential equations in positive characteristic.

Abstract: Linear differential equations form the central topic of this volume, Galois theory being the unifying theme.
A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used.
This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

942 citations

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02 Nov 2007

TL;DR: In a course given by the author at Harvard University in the fall semester of 1988 as mentioned in this paper, the course focused on the inverse problem of Galois Theory: the construction of field extensions having a given finite group as Galois group.

Abstract: This book is based on a course given by the author at Harvard University in the fall semester of 1988. The course focused on the inverse problem of Galois Theory: the construction of field extensions having a given finite group as Galois group. In the first part of the book, classical methods and results, such as the Scholz and Reichardt construction for p-groups, p != 2, as well as Hilbert's irreducibility theorem and the large sieve inequality, are presented. The second half is devoted to rationality and rigidity criteria and their application in realizing certain groups as Galois groups of regular extensions of Q(T). While proofs are not carried out in full detail, the book contains a number of examples, exercises, and open problems.

528 citations

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TL;DR: In this paper, the authors extend the results of [CHT] by removing the minimal ramification condition on the lifts and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge-Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension.

Abstract: We extend the results of [CHT] by removing the ‘minimal ramification’ condition on the lifts. That is we establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension. The main innovation is a new approach to the automorphy of non-minimal lifts which is closer in spirit to the methods of [TW] than to those of [W], which relied on Ihara’s lemma.

478 citations