Galois theory is a central part of algebra, dealing with symmetries between solutions of algebraic equations in one variable. This collection of papers brings together articles from some of the world's leading experts in this field. Topics center around the Inverse Galois Problem, comprising the full range of methods and approaches in this area, making this an invaluable resource for all those whose research involves Galois theory.
Author(s): Helmut Voelklein (editor), J. G. Thompson (editor), David Harbater (editor), Peter Müller (editor)
Series: London Mathematical Society Lecture Note Series 256
Publisher: Cambridge University Press
Year: 1999
Language: English
Commentary: OCR'd with ABBYY Finereader (not proofread)
Pages: 292
Aspects of Galois Theory
CONTENTS
INTRODUCTION
GALOIS THEORY OF SEMILINEAR TRANSFORMATIONS by Shreeram S. Abhyankar
Section 1: Introduction
Section 2: Linear Groups
Section 3: Iterated Linear Groups
Section 4: Symplectic Groups
REFERENCES
Tools for the computation of families of coverings by Jean-Marc Couveignes
1 Introduction
2 Two coverings ramified over three points
3 Topological description
4 Patching
5 Looking for algebraic dependencies
5.1 General procedure
5.2 Using numerical approximations
6 Stable curves
7 Conclusion
References
1 Rational Geometric Fundamental Groups
2 Abelian coverings
3 Coverings of P1 with given ramification type
3.1 The Construction Principle
3.2 Rigidity
4 Construction of infinite towers of unrami¬
fied curve covers
4.1 Thompson tuples and Belyi triples
4#2 Curves with infinite towers of unramified regular Galois extensions
4.3 Geometric Interpretation of Thompson tuples
5 A family of curves with infinite geometric fundamental group
5.1 The abelian variety J^ew
5.2 Rational points on A (P(SJ)
5.3 Example
References
Some Arithmetic Properties of Algebraic Covers by Pierre Debes
1. Introduction
2. Structure result for models of a cover
3. Covers with prescribed fibers
4. Field of moduli and extension of constants
5. The local-to-global principle
References
Curves with infinite A-rat ion al geometric fundamental group by Gerhard Frey, Ernst Kani and Helmut Vdlklein
1 Rational Geometric Fundamental Groups
2 Abelian coverings
3 Coverings of P1 with given ramification type
3.1 The Construction Principle
3.2 Rigidity
4 Construction of infinite towers of unramified curve covers
4.1 Thompson tuples and Belyi triples
4.2 Curves with infinite towers of unramified regular Galois extensions
4.3 Geometric Interpretation of Thompson tuples
5 A family of curves with infinite geometric fundamental group
5.1 The abelian variety J^new/C
5.2 Rational points on A (P(Si)
5.3 Example
References
Embedding Problems and Adding Branch Points by David Harbater
Section 1. Introduction and survey of results.
Section 2. Notions concerning covers and groups.
Section 3. Results via patching.
Section 4. Results via lifting.
Section 5. The main result.
References[FJ]
On beta and gamma functions associated with the Grothendieck-Teichmiiller groups by Yasutaka Ihara
Introduction
1 Statement of main results and discussions
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
2 Proofs of the main results
2.1
2.2
2.3
2.4
2.5
2.6
3 Appendix
3.1
3.2
3.3 Proof of Theorem A-l
3.4 Proof of Theorem A-2
References
Arithmetically exceptional functions and elliptic curves by Peter Muller
1 Introduction
2 Arithmetically exceptional rational functions
3 Branch cycle descriptions in geometric mo- nodromy groups
4 Monodromy groups of arithmetically exceptional functions
5 Rational functions with branching type
5.1 Rational isogenies of degree 5
6 Application to a question of Thompson
6.1 Thompson’s question, group-theoretic preparation
6.2 Passing to a different rational function
6.3 Rationality question for |A/G| <_ 2
6.4 Existence of f for |A/G| < 2
A Computation of the (2, 2, 2,4)-example
References
Tangential base points and Eisenstein power series by Hiroaki Nakamura
I
II
III
References
Braid-abelian tuples in Spn(K) by John Thompson and Helmut Volklein University of Florida
0 Introduction
1 Triples of bi-perspectivities in GL4(K)
2 The tuples in Spn(K)
3 Braiding action on the tuples
4 Existence of the tuples, and generated subgroup
5 The resulting Galois realizations
References
Deformation of tame admissible covers of curves by Stefan. Wewers
Introduction
The main result
Outline
1 Formal patching
1.1 Outline of the proof
1.2 Etale localization
1.3 Descent
1.4 Grothendieck’s Existence Theorem
2 Tame admissible covers
2.1 Formal double points
2.2 Marked nodal curves
2.3 Tame admissible covers
3 Deformation theory
3.1 Statement of the main result
3.2 Proof of the main theorem
4 Tame fundamental groups of smooth curves
4.1 The tame fundamental group as a Galois group
4.2 Tame covers over complete discrete valuation rings
4.3 The specialization morphism
5 Appendix
References
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