Aspects of Galois Theory

Description

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. 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|