**Proceedings of Symposia in Pure Mathematics**

Volume: 70;
2002;
569 pp;
Hardcover

MSC: Primary 20; 14; 12; 11; 16;

Print ISBN: 978-0-8218-2036-0

Product Code: PSPUM/70

List Price: $144.00

Individual Member Price: $115.20

**Electronic ISBN: 978-0-8218-9375-3
Product Code: PSPUM/70.E**

List Price: $144.00

Individual Member Price: $115.20

# Arithmetic Fundamental Groups and Noncommutative Algebra

Share this page *Edited by *
*Michael D. Fried; Yasutaka Ihara*

The arithmetic and geometry of moduli spaces and their fundamental groups are a very active research area. This book offers a complete overview of developments made over the last decade.

The papers in this volume examine the geometry of moduli spaces of curves with a function on them. The main players in Part 1 are the absolute Galois group \(G_{\mathbb Q}\) of the algebraic numbers and its close relatives. By analyzing how \(G_{\mathbb Q}\) acts on fundamental groups defined by Hurwitz moduli problems, the authors achieve a grand generalization of Serre's program from the 1960s.

Papers in Part 2 apply \(\theta\)-functions and configuration spaces to the study of fundamental groups over positive characteristic fields. In this section, several authors use Grothendieck's famous lifting results to give extensions to wildly ramified covers. Properties of the fundamental groups have brought collaborations between geometers and group theorists. Several Part 3 papers investigate new versions of the genus 0 problem. In particular, this includes results severely limiting possible monodromy groups of sphere covers. Finally, Part 4 papers treat Deligne's theory of Tannakian categories and arithmetic versions of the Kodaira-Spencer map.

This volume is geared toward graduate students and research mathematicians interested in arithmetic algebraic geometry.

#### Table of Contents

# Table of Contents

## Arithmetic Fundamental Groups and Noncommutative Algebra

- Contents v6 free
- Prelude: Arithmetic fundamental groups and noncommutative algebra vii8 free
- Part 1. G[sub(Q)] action on moduli spaces of covers 132 free
- Descent theory for algebraic covers 334
- Galois invariants of dessins d'enfants 2758
- Limits of Galois representations in fundamental groups along maximal degeneration of marked curves, II 4374
- Hurwitz monodromy, spin separation and higher levels of a modular tower 79110
- Field of moduli and field of definition of Galois covers 221252
- Some arithmetic aspects of Galois actions on the pro-p fundamental group of P[sup(1)] … {0, l, ∞} 247278
- Relationships between conjectures on the structure of pro-p Galois groups unramified outside p 275306
- On explicit formulae for l-adic polylogarithms 285316

- Part 2. Curve covers in positive characteristic 295326
- Part 3. Special groups for covers of the punctured sphere 407438
- Part 4. Fundamental groupoids and Tannakian categories 495526

#### Readership

Graduate students and research mathematicians interested in arithmetic algebraic geometry.