The Grothendieck Theory of Dessins d'Enfants

by Leila Schneps

• ISBN13: 9780521478212
• ISBN10: 0521478219
• Format: Paperback
• Copyright: 1994-09-30
• Publisher: Cambridge University Press

Summary

Dessins d'Enfants are combinatorial objects, namely drawings with vertices and edges on topological surfaces. Their interest lies in their relation with the set of algebraic curves defined over the closure of the rationals, and the corresponding action of the absolute Galois group on them. The study of this group via such realted combinatorial methods as its action on the Dessins and on certain fundamental groups of moduli spaces was initiated by Alexander Grothendieck in his unpublished Esquisse d'un Programme, and developed by many of the mathematicians who have contributed to this volume. The various articles here unite all of the basics of the subject as well as the most recent advances. Researchers in number theory, algebraic geometry or related areas of group theory will find much of interest in this book.

Table of Contents

1. Noncongruence subgroups, covers, and drawings B. Birch

2. Dessins d'enfant on the Riemann sphere L. Schneps

3. Dessins from a geometric point of view J-M. Couveignes and L. Granboulan

4. Maps, hypermaps and triangle groups G. Jones and D. Singerman

5. Fields of definition of some three point ramified field extensions G. Malle

6. On the classification of plane trees by their Galois orbit G. Shabat

7. Triangulations M. Bauer and C. Itzykson

8. Dessins d'enfant and Shimura varieties P. Cohen

9. Horizontal divisors on arithmetic surfaces associated with Belyi uniformizations Y. Ihara

10. Algebraic representation of the Teichmü

ller spaces K. Saito

11. On the embedding of Gal(Q/Q) into GT Y. Ihara

Appendix M. Emsalem and P. Lochak

12. The Grothendieck-Teichmü

ller group and automorphisms of braid groups P. Lochak and L. Schneps

13. Moore and Seiberg equations, topological field theories and Galois theory P. Degiovanni.