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Geometric Galois Actions. Volume 2. The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups. / edited by Leila Schneps, Pierre Lochak.

This book surveys progress in the domains described in the hitherto unpublished manuscript 'Esquisse d'un Programme' (Sketch of a Program) by Alexander Grothendieck. It will be of wide interest amongst workers in algebraic geometry, number theory, algebra and topology.

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Bibliographic Details
Online Access: Full Text (via Cambridge)
Other Authors: Schneps, Leila, Lochak, P. (Pierre)
Other title:Inverse Galois Problem, Moduli Spaces and Mapping Class group
Format: eBook
Language:English
Published: Cambridge : Cambridge University Press, 1997.
Series:London Mathematical Society lecture note series ; no. 243.
Subjects:
Geometry, Algebraic.
Moduli theory.
Geometry, Algebraic
Moduli theory
Table of Contents:
  • Cover
  • Title
  • Copyright
  • Contents
  • Introduction
  • Abstracts of the talks
  • Short Courses
  • Individual talks
  • Evening seminar on Teichmiiller and moduli space.
  • Part I. Dessins d'enfants
  • Unicellular Cartography and Galois Orbits of Plane Trees
  • 0. Introduction
  • 1. Belyi theorem for unicellular dessins
  • 2. Edge rotation groups of unicellular dessins
  • 3. Combinatorial structures help to see the edge rotation groups
  • 4. Generalized Chebyshev polynomials of cartographically special trees
  • References
  • Galois Groups, Monodromy Groups and Cartographic Groups.
  • 0. Introduction
  • 1. The absolute Galois group
  • 2. Belyfs Theorem
  • 3. Belyt functions and dessins
  • 4. Belyi pairs and permutations
  • 5. Belyi's Theorem and uniformisation
  • 6. Plane trees and Shabat polynomials
  • 7. Examples of plane trees and their groups
  • Appendix
  • Permutation techniques for coset representations of modular subgroups
  • 1. Introduction
  • 2. Identifying congruence subgroups
  • 3. Enlarging subgroups
  • 4. Remarks and acknowledgements
  • References
  • Dessins d'enfants en genre 1
  • 0. Introduction.
  • 1. Un exemple parametrique.
  • 2. Dessins en genre 1, points de torsion et formes modulaires.
  • 3. Dessins d'enfants en genre 1 et isogenies. Un exemple.
  • 4 Remerciements.
  • References
  • Part II. The Inverse Galois Problem
  • The Regular Inverse Galois Problem over Large Fields
  • 1. Introduction.
  • 2. Conjectures.
  • 3. Results.
  • 4 Main arguments.
  • References
  • The Symplectic Braid Group and Galois Realizations
  • 0. Introduction
  • 1. Artin's braid group
  • 2. Coverings
  • 3. Varieties associated with the Coxeter group of type Ct
  • 4. Choosing the group G.
  • 5. Generators of the symplectic braid group
  • References
  • Applying Modular Towers to the Inverse Galois Problem
  • 0. Introduction to the main problem.
  • 1. Precise versions of the main conjecture.
  • 2. Construction of universal Prattini covers.
  • 3. Progress on the case A5 and C = C3r.
  • 4.A. Lifting elements of order p.
  • Appendix I. Nielsen classes and Modular Towers.
  • Appendix II. Equivalence of covers of the sphere
  • References
  • Part III. Galois actions and mapping class groups
  • Galois group GQ, Singularity E7, and Moduli M3
  • 0. Introduction
  • 1. E7 and M3.
  • 2. Tangential morphisms
  • 3. Galois action on Artin groups
  • References
  • Monodromy of Iterated Integrals and Non-abelian Unipotent Periods
  • 0. Introduction.
  • 1. Canonical connection with logarithmic singularities.
  • 2. The Gauss-Manin connection associated with the morphism
  • 3. Homotopy relative tangential base points on P1 (C)\{a1 ..., an+1}.
  • 4. Generators of 7Ti(P1(C)\{a1 ..., an+1}, x).
  • 5. Monodromy of iterated integrals on P1(C)\{a1 ..., an+1}
  • 6. Calculations.
  • 7. Configuration spaces.
  • 8. The Drinfeld-Ihara Z/5-cycle relation.

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