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Arithmetic Fundamental Groups and Noncommutative Algebra
Edited by: Michael D. Fried, University of California, Irvine, CA, and Yasutaka Ihara, RIMS, Kyoto University, Japan
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Proceedings of Symposia in Pure Mathematics
2002; 569 pp; hardcover
Volume: 70
ISBN-10: 0-8218-2036-2
ISBN-13: 978-0-8218-2036-0
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 Kodaira-Spencer map.

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

Graduate students and research mathematicians interested in arithmetic algebraic geometry.

$$G_{\mathbb Q}$$ 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 Hodge-Arakelov theory of elliptic curves I