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Arithmetic Fundamental Groups and Noncommutative Algebra

Edited by: Michael D. Fried University of California, Irvine, CA
Yasutaka Ihara RIMS, Kyoto University, Kyoto, Japan
Available Formats:
Hardcover ISBN: 978-0-8218-2036-0
Product Code: PSPUM/70
569 pp
List Price: $144.00 MAA Member Price:$129.60
AMS Member Price: $115.20 Electronic ISBN: 978-0-8218-9375-3 Product Code: PSPUM/70.E 569 pp List Price:$144.00
MAA Member Price: $129.60 AMS Member Price:$115.20
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List Price: $216.00 MAA Member Price:$194.40
AMS Member Price: $172.80 Click above image for expanded view Arithmetic Fundamental Groups and Noncommutative Algebra Edited by: Michael D. Fried University of California, Irvine, CA Yasutaka Ihara RIMS, Kyoto University, Kyoto, Japan Available Formats:  Hardcover ISBN: 978-0-8218-2036-0 Product Code: PSPUM/70 569 pp  List Price:$144.00 MAA Member Price: $129.60 AMS Member Price:$115.20
 Electronic ISBN: 978-0-8218-9375-3 Product Code: PSPUM/70.E 569 pp
 List Price: $144.00 MAA Member Price:$129.60 AMS Member Price: $115.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$216.00
MAA Member Price: $194.40 AMS Member Price:$172.80
• Book Details

Proceedings of Symposia in Pure Mathematics
Volume: 702002
MSC: Primary 20; 14; 12; 11; 16;

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.

• Part 1: $G_{\mathbb {Q}}$ action on moduli spaces of covers
• Pierre Dèbes - Descent theory for algebraic covers [ MR 1935403 ]
• Jordan S. Ellenberg - Galois invariants of dessins d’enfants [ MR 1935404 ]
• Hiroaki Nakamura - Limits of Galois representations in fundamental groups along maximal degeneration of marked curves. II [ MR 1935405 ]
• Paul Bailey and Michael D. Fried - Hurwitz monodromy, spin separation and higher levels of a modular tower [ MR 1935406 ]
• Stefan Wewers - Field of moduli and field of definition of Galois covers [ MR 1935407 ]
• Yasutaka Ihara - Some arithmetic aspects of Galois actions in the pro-$p$ fundamental group of $\mathbb {P}^1-\{0,1,\infty \}$ [ MR 1935408 ]
• Romyar T. Sharifi - Relationships between conjectures on the structure of pro-$p$ Galois groups unramified outside $p$ [ MR 1935409 ]
• Hiroaki Nakamura and Zdzisław Wojtkowiak - On explicit formulae for $l$-adic polylogarithms [ MR 1935410 ]
• Part 2. Curve covers in positive characteristic
• Akio Tamagawa - Fundamental groups and geometry of curves in positive characteristic [ MR 1935411 ]
• Michel Raynaud - Sur le groupe fondamental d’une courbe complète en caractéristique $p > 0$ [ MR 1935412 ]
• Michael D. Fried and Ariane Mézard - Configuration spaces for wildly ramified covers [ MR 1935413 ]
• Marco A. Garuti - Linear systems attached to cyclic inertia [ MR 1935414 ]
• Robert Guralnick and Katherine F. Stevenson - Prescribing ramification [ MR 1935415 ]
• Part 3. Specials groups for covers of the punctured sphere
• Shreeram S. Abhyankar - Desingularization and modular Galois theory [ MR 1935416 ]
• Dan Frohardt, Robert Guralnick and Kay Magaard - Genus 0 actions of groups of Lie rank 1 [ MR 1935417 ]
• Helmut Völklein - Galois realizations of profinite projective linear groups [ MR 1935418 ]
• Part 4. Fundamental groupoids and Tannakian categories
• Shlomo Gelaki - Semisimple triangular Hopf algebras and Tannakian categories [ MR 1935419 ]
• Phùng Hồ Hải - On a theorem of Deligne on characterization of Tannakian categories [ MR 1935420 ]
• Shinichi Mochizuki - A survey of the Hodge-Arakelov theory of elliptic curves I [ MR 1935421 ]
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Volume: 702002
MSC: Primary 20; 14; 12; 11; 16;

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.

• Part 1: $G_{\mathbb {Q}}$ action on moduli spaces of covers
• Pierre Dèbes - Descent theory for algebraic covers [ MR 1935403 ]
• Jordan S. Ellenberg - Galois invariants of dessins d’enfants [ MR 1935404 ]
• Hiroaki Nakamura - Limits of Galois representations in fundamental groups along maximal degeneration of marked curves. II [ MR 1935405 ]
• Paul Bailey and Michael D. Fried - Hurwitz monodromy, spin separation and higher levels of a modular tower [ MR 1935406 ]
• Stefan Wewers - Field of moduli and field of definition of Galois covers [ MR 1935407 ]
• Yasutaka Ihara - Some arithmetic aspects of Galois actions in the pro-$p$ fundamental group of $\mathbb {P}^1-\{0,1,\infty \}$ [ MR 1935408 ]
• Romyar T. Sharifi - Relationships between conjectures on the structure of pro-$p$ Galois groups unramified outside $p$ [ MR 1935409 ]
• Hiroaki Nakamura and Zdzisław Wojtkowiak - On explicit formulae for $l$-adic polylogarithms [ MR 1935410 ]
• Part 2. Curve covers in positive characteristic
• Akio Tamagawa - Fundamental groups and geometry of curves in positive characteristic [ MR 1935411 ]
• Michel Raynaud - Sur le groupe fondamental d’une courbe complète en caractéristique $p > 0$ [ MR 1935412 ]
• Michael D. Fried and Ariane Mézard - Configuration spaces for wildly ramified covers [ MR 1935413 ]
• Marco A. Garuti - Linear systems attached to cyclic inertia [ MR 1935414 ]
• Robert Guralnick and Katherine F. Stevenson - Prescribing ramification [ MR 1935415 ]
• Part 3. Specials groups for covers of the punctured sphere
• Shreeram S. Abhyankar - Desingularization and modular Galois theory [ MR 1935416 ]
• Dan Frohardt, Robert Guralnick and Kay Magaard - Genus 0 actions of groups of Lie rank 1 [ MR 1935417 ]
• Helmut Völklein - Galois realizations of profinite projective linear groups [ MR 1935418 ]
• Part 4. Fundamental groupoids and Tannakian categories
• Shlomo Gelaki - Semisimple triangular Hopf algebras and Tannakian categories [ MR 1935419 ]
• Phùng Hồ Hải - On a theorem of Deligne on characterization of Tannakian categories [ MR 1935420 ]
• Shinichi Mochizuki - A survey of the Hodge-Arakelov theory of elliptic curves I [ MR 1935421 ]
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