Arithmetic Fundamental Groups and Noncommutative Algebra: 1999 Von Neumann Conference on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16-27, 1999, Mathematical Sciences Research Institut(Proceedings of Symposia in Pure Mathematics)

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:
$G_{\mathbb Q}$ action on moduli spaces of covers: Descent theory for algebraic covers by P. Debes Galois invariants of dessins d'enfants by J. S. Ellenberg Limits of Galois representations in fundamental groups along maximal degeneration of marked curves, II by H. Nakamura Hurwitz monodromy, spin separation and higher levels of a modular tower by P. Bailey and M. D. Fried Field of moduli and field of definition of Galois covers by S. Wewers Some arithmetic aspects of Galois actions on the pro-$p$ fundamental group of ${\mathbb P}^1-\{0,1,\infty\}$ by Y. Ihara Relationships between conjectures on the structure of pro-$p$ Galois groups unramified outside $p$ by R. T. Sharifi On explicit formulae for $l$-adic polylogarithms by H. Nakamura and Z. Wojtkowiak Curve covers in positive characteristic: Fundamental groups and geometry of curves in positive characteristic by A. Tamagawa Sur le groupe fondamental d'une courbe complete en caracteristique $p>0$ by M. Raynaud Configuration spaces for wildly ramified covers by M. D. Fried and A. Mezard Linear systems attached to cyclic inertia by M. A. Garuti Prescribing ramification by R. Guralnick and K. F. Stevenson Special groups for covers of the punctured sphere: Desingularization and modular Galois theory by S. S. Abhyankar and D. Harbater Genus 0 actions of groups of Lie rank 1 by D. Frohardt, R. Guralnick, and K. Magaard Galois realizations of profinite projective linear groups by H. Volklein Fundamental groupoids and Tannakian categories: Semisimple triangular Hopf algebras and Tannakian categories by S. Gelaki On a theorem of Deligne on characterization of Tannakian categories by P. H. Hai A survey of the Hodge-Arakelov theory of elliptic curves I by S. Mochizuki.