Proceedings of Symposia in Pure Mathematics 2002; 569 pp; hardcover Volume: 70 ISBN10: 0821820362 ISBN13: 9780821820360 List Price: US$144 Member Price: US$115.20 Order Code: PSPUM/70
 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 KodairaSpencer map. This volume is geared toward graduate students and research mathematicians interested in arithmetic algebraic geometry. Readership Graduate students and research mathematicians interested in arithmetic algebraic geometry. Table of Contents \(G_{\mathbb Q}\) action on moduli spaces of covers  P. Dèbes  Descent theory for algebraic covers
 J. S. Ellenberg  Galois invariants of dessins d'enfants
 H. Nakamura  Limits of Galois representations in fundamental groups along maximal degeneration of marked curves, II
 P. Bailey and M. D. Fried  Hurwitz monodromy, spin separation and higher levels of a modular tower
 S. Wewers  Field of moduli and field of definition of Galois covers
 Y. Ihara  Some arithmetic aspects of Galois actions on the pro\(p\) fundamental group of \({\mathbb P}^1\{0,1,\infty\}\)
 R. T. Sharifi  Relationships between conjectures on the structure of pro\(p\) Galois groups unramified outside \(p\)
 H. Nakamura and Z. Wojtkowiak  On explicit formulae for \(l\)adic polylogarithms
Curve covers in positive characteristic  A. Tamagawa  Fundamental groups and geometry of curves in positive characteristic
 M. Raynaud  Sur le groupe fondamental d'une courbe complète en caractéristique \(p>0\)
 M. D. Fried and A. Mézard  Configuration spaces for wildly ramified covers
 M. A. Garuti  Linear systems attached to cyclic inertia
 R. Guralnick and K. F. Stevenson  Prescribing ramification
Special groups for covers of the punctured sphere  S. S. Abhyankar and D. Harbater  Desingularization and modular Galois theory
 D. Frohardt, R. Guralnick, and K. Magaard  Genus 0 actions of groups of Lie rank 1
 H. Völklein  Galois realizations of profinite projective linear groups
Fundamental groupoids and Tannakian categories  S. Gelaki  Semisimple triangular Hopf algebras and Tannakian categories
 P. H. Hai  On a theorem of Deligne on characterization of Tannakian categories
 S. Mochizuki  A survey of the HodgeArakelov theory of elliptic curves I
