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Table of Contents
Intro; Preface; Contents; I The Rigidity Method; 1 The Inverse Galois Problem over C(t) and R(t); 1.1 The Fundamental Group of the Punctured RiemannSphere; 1.2 The Algebraic Variant of the Fundamental Group; 1.3 Extension by Complex Conjugation; 1.4 Generalization to Function Fields of Riemann Surfaces; 2 Arithmetic Fundamental Groups; 2.1 Descent to Algebraically Closed Subfields; 2.2 The Fundamental Splitting Sequence; 2.3 The Action via the Cyclotomic Character; 2.4 The Theorem of Belyi; 3 Fields of Definition of Galois Extensions; 3.1 Cyclic and Projective Descent
3.2 Fields of Definition of Geometric Field Extensions3.3 Fields of Definition of Geometric Galois Extensions; 4 The Rigidity Property; 4.1 The Hurwitz Classification; 4.2 The Fixed Field of a Class of Generating Systems; 4.3 The Basic Rigidity Theorem; 4.4 Choice of Ramification Points; 5 Verification of Rigidity; 5.1 Geometric Galois Extensions over Q(t)with Abelian Groups; 5.2 Geometric Galois Extensions over Q(t) with Sn and An; 5.3 Structure Constants; 5.4 The Rigidity Criterion of Belyi; 6 Geometric Automorphisms; 6.1 Extension of the Algebraic Fundamental Group
6.2 The Action of Geometric Automorphisms6.3 Rigid Orbits; 6.4 The Twisted Rigidity Theorem; 6.5 Geometric Galois Extensions over Q(t) with M12and M11; 7 Rational Translates of Galois Extensions; 7.1 Galois Rational Translates; 7.2 Rational Translates with Few Ramification Points; 7.3 Twisting Rational Translates; 7.4 Geometric Galois Extensions over Q(t) with L2(p); 8 Automorphisms of the Galois Group; 8.1 Fixed Fields of Coarse Classes of Generating Systems; 8.2 Extension of the Galois Group by Outer Automorphisms; 8.3 Geometric Extension of the Galois Group by Outer Automorphisms
8.4 Geometric Galois Extensions over Q(t) with PGL2(p)9 Computation of Polynomials with Prescribed Group; 9.1 Decomposition of Prime Divisors in Galois Extensions; 9.2 Polynomials with Groups Sn and An; 9.3 Polynomials with the Group Aut(A6) and RelatedGroups; 9.4 Polynomials with the Mathieu Groups M12 and M11; 10 Specialization of Geometric Galois Extensions; 10.1 Local Structure Stability; 10.2 Reality Questions; 10.3 Ramification in Minimal Fields of Definition; 10.4 Ramification in Residue Fields; II Applications of Rigidity; 1 The General Linear Groups; 1.1 Groups of Lie Type
1.2 Rigidity for GLn(q)1.3 Galois Realizations for Linear Groups; 2 Pseudo-Reflection Groups and Belyi Triples; 2.1 Groups Generated by Pseudo-Reflections; 2.2 An Effective Version of Belyi's Criterion; 2.3 Imprimitive and Symmetric Groups; 2.4 Invariant Forms; 3 The Classical Groups; 3.1 Rigidity for GUn(q); 3.2 Rigidity for CSp2n(q); 3.3 Rigidity for SO2n+1(q); 3.4 Rigidity for CO2n+(q); 3.5 Rigidity for CO2n-(q); 4 The Exceptional Groups of Rank at Most 2; 4.1 Divisibility Criteria; 4.2 Rigidity for the Ree Groups 2G2(q2); 4.3 Rigidity for the Groups G2(q)
3.2 Fields of Definition of Geometric Field Extensions3.3 Fields of Definition of Geometric Galois Extensions; 4 The Rigidity Property; 4.1 The Hurwitz Classification; 4.2 The Fixed Field of a Class of Generating Systems; 4.3 The Basic Rigidity Theorem; 4.4 Choice of Ramification Points; 5 Verification of Rigidity; 5.1 Geometric Galois Extensions over Q(t)with Abelian Groups; 5.2 Geometric Galois Extensions over Q(t) with Sn and An; 5.3 Structure Constants; 5.4 The Rigidity Criterion of Belyi; 6 Geometric Automorphisms; 6.1 Extension of the Algebraic Fundamental Group
6.2 The Action of Geometric Automorphisms6.3 Rigid Orbits; 6.4 The Twisted Rigidity Theorem; 6.5 Geometric Galois Extensions over Q(t) with M12and M11; 7 Rational Translates of Galois Extensions; 7.1 Galois Rational Translates; 7.2 Rational Translates with Few Ramification Points; 7.3 Twisting Rational Translates; 7.4 Geometric Galois Extensions over Q(t) with L2(p); 8 Automorphisms of the Galois Group; 8.1 Fixed Fields of Coarse Classes of Generating Systems; 8.2 Extension of the Galois Group by Outer Automorphisms; 8.3 Geometric Extension of the Galois Group by Outer Automorphisms
8.4 Geometric Galois Extensions over Q(t) with PGL2(p)9 Computation of Polynomials with Prescribed Group; 9.1 Decomposition of Prime Divisors in Galois Extensions; 9.2 Polynomials with Groups Sn and An; 9.3 Polynomials with the Group Aut(A6) and RelatedGroups; 9.4 Polynomials with the Mathieu Groups M12 and M11; 10 Specialization of Geometric Galois Extensions; 10.1 Local Structure Stability; 10.2 Reality Questions; 10.3 Ramification in Minimal Fields of Definition; 10.4 Ramification in Residue Fields; II Applications of Rigidity; 1 The General Linear Groups; 1.1 Groups of Lie Type
1.2 Rigidity for GLn(q)1.3 Galois Realizations for Linear Groups; 2 Pseudo-Reflection Groups and Belyi Triples; 2.1 Groups Generated by Pseudo-Reflections; 2.2 An Effective Version of Belyi's Criterion; 2.3 Imprimitive and Symmetric Groups; 2.4 Invariant Forms; 3 The Classical Groups; 3.1 Rigidity for GUn(q); 3.2 Rigidity for CSp2n(q); 3.3 Rigidity for SO2n+1(q); 3.4 Rigidity for CO2n+(q); 3.5 Rigidity for CO2n-(q); 4 The Exceptional Groups of Rank at Most 2; 4.1 Divisibility Criteria; 4.2 Rigidity for the Ree Groups 2G2(q2); 4.3 Rigidity for the Groups G2(q)