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Book DetailsProceedings of Symposia in Pure MathematicsVolume: 70; 2002; 569 ppMSC: 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 KodairaSpencer map.
This volume is geared toward graduate students and research mathematicians interested in arithmetic algebraic geometry.ReadershipGraduate students and research mathematicians interested in arithmetic algebraic geometry.

Table of Contents

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 HodgeArakelov theory of elliptic curves I [ MR 1935421 ]


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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.
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 HodgeArakelov theory of elliptic curves I [ MR 1935421 ]