## 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)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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### 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 |

### Common terms and phrases

abelian affine arithmetic assume automorphism braid group branch cycles branch locus branch points characteristic cohomology complex conjugation components compute Conjecture conjugacy classes consider corresponding cusps defined Dehn twists denote dessin divisor element elliptic curve equivalence etale extension fiber field of definition field of moduli finite group fixed Frattini functions functor fundamental group G-cover Galois closure Galois cover Galois group genus geometric gerbe gives group G H-M reps homomorphism Hopf algebras Hurwitz family Hurwitz space induces inertia group integer Inverse Galois Problem irreducible isomorphism kernel Lemma Lie algebra Math modular curves Modular Tower moduli space monodromy group morphism Nielsen class nontrivial notation obstruction orbits p-Frattini cover permutation prime pro-p profinite group PROOF Prop PROPOSITION quotient ramification real points representation resp result sequence spin separating subgroup Suppose surjective tangential base point Theorem theory trivial unramified