# Arithmetic Fundamental Groups and Noncommutative Algebra: 1999 Von Neumann Conference on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16-27, 1999, Mathematical Sciences Research Institute, Berkeley, California (Google eBook)

Michael D. Fried, Yasutaka Ihara
American Mathematical Soc. - Mathematics - 569 pages
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.

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### Contents

 Galois invariants of dessins denfants 27 Limits of Galois representations in fundamental groups along maximal 43 Hurwitz monodromy spin separation and higher levels of a modular tower 79 Field of moduli and field of definition of Galois covers 220 Some arithmetic aspects of Galois actions on the prop fundamental group 247 Relationships between conjectures on the structure of prop Galois groups 273 Fundamental groups and geometry of curves in positive characteristic 297 Sur le groupe fondamental dune courbe complete en caracteristique p 0 335
 Configuration spaces for wildly ramified covers 353 MARCO A GARUTI 377 Desingularization and modular Galois theory 409 actions of groups of Lie rank 1 449 Galois realizations of profinite projective linear groups 485 On a theorem of Deligne on characterization of Tannakian categories 517 A survey of the HodgeA rake lov theory of elliptic curves I 533 Copyright