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Arithmetic Fundamental Groups and Noncommutative Algebra Proceedings of Symposia in PURE MATHEMATICS

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Arithmetic Fundamental Groups and Noncommutative Algebra Proceedings of Symposia in PURE MATHEMATICS http://dx.doi.org/10.1090/pspum/070 Arithmetic Fundamental Groups and Noncommutative Algebra Proceedings of Symposia in PURE MATHEMATICS Volume 70 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 Michael D. Fried Yasutaka Ihara Editors PROCEEDINGS OF THE 1999 VON NEUMANN CONFERENCE ON ARITHMETIC FUNDAMENTAL GROUPS AND NONCOMMUTATIVE ALGEBRA, HELD AT THE MATHEMATICAL SCIENCES RESEARCH INSTITUTE, BERKELEY, CALIFORNIA, AUGUST 16-27, 1999 with support from the National Security Agency, Grant MDA904-99-1-0062. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Security Agency. 2000 Mathematics Subject Classification. Primary 20F34, 14E20, 14F35, 12F12, 20C15, 20D06, 14E20, 14H30, 11R32, 16W30. Library of Congress Cataloging-in-Publication Data Von Neumann Conference on Arithmetic Fundamental Groups and Noncommutative Algebra (1999 : Berkeley, Calif.) Arithmetic fundamental groups and noncommutative algebra : 1999 Von Neumann Conference on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16-27, 1999, Math• ematical Sciences Research Institute, Berkeley, California / Michael D. Fried, Yasutaka Ihara, editors. p. cm. — (Proceedings of symposia in pure mathematics, ISSN 0082-0717 ; V. 70) Includes bibliographical references. ISBN 0-8218-2036-2 (alk. paper) 1. Fundamental groups (Mathematics)—Congresses. 2. Noncommutative algebra—Congresses. I. Fried, Michael D., 1942- II. Ihara, Y. (Yasutaka), 1938- III. Title. IV. Series. QA177.V66 1999 512'.7—dc21 2002021586 Copying and reprinting. Material in this book may be reproduced by any means for edu• cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg• ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math• ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02 Contents Prelude: Arithmetic fundamental groups and noncommutative algebra vii Part 1. GQ action on moduli spaces of covers 1 Descent theory for algebraic covers PIERRE DEBES 3 Galois invariants of dessins d'enfants JORDAN S. ELLENBERG 27 Limits of Galois representations in fundamental groups along maximal degeneration of marked curves, II HIROAKI NAKAMURA 43 Hurwitz monodromy, spin separation and higher levels of a modular tower PAUL BAILEY AND MICHAEL D. FRIED 79 Field of moduli and field of definition of Galois covers STEFAN WEWERS 221 Some arithmetic aspects of Galois actions on the pro-p fundamental group ofP1 -{0,l,oo} YASUTAKA IHARA 247 Relationships between conjectures on the structure of pro-p Galois groups unramified outside p ROMYAR T. SHARIFI 275 On explicit formulae for /-adic polylogarithms HIROAKI NAKAMURA AND ZDZISLAW WOJTKOWIAK 285 Part 2. Curve covers in positive characteristic 295 Fundamental groups and geometry of curves in positive characteristic AKIO TAMAGAWA 297 Sur le groupe fondamental d'une courbe complete en caracteristique p > 0 MICHEL RAYNAUD 335 Configuration spaces for wildly ramified covers MICHAEL D. FRIED AND ARIANE MEZARD 353 vi CONTENTS Linear systems attached to cyclic inertia MARCO A. GARUTI 377 Prescribing ramification ROBERT GURALNICK AND KATHERINE F. STEVENSON 387 Part 3. Special groups for covers of the punctured sphere 407 Desingularization and modular Galois theory SHREERAM S. ABHYANKAR with an appendix by DAVID HARBATER 409 Genus 0 actions of groups of Lie rank 1 DAN FROHARDT, ROBERT GURALNICK, AND KAY MAGAARD 449 Galois realizations of profinite projective linear groups HELMUT VOLKLEIN 485 Part 4. Fundamental groupoids and Tannakian categories 495 Semisimple triangular Hopf algebras and Tannakian categories SHLOMO GELAKI 497 On a theorem of Deligne on characterization of Tannakian categories PHUNG HO HAI 517 A survey of the Hodge-Arakelov theory of elliptic curves I SHINICHI MOCHIZUKI 533 Proceedings of Symposia in Pure Mathematics Volume 70. 2002 Prelude: Arithmetic Fundamental Groups and Noncommutative Algebra Michael D. Fried ABSTRACT. From number theory to string theory, analyzing algebraic relations in two variables still dominates how we view laws governing relations between quantities. An algebraic relation between two variables defines a nonsingular projective curve. Our understanding starts with moduli of curves. From, however, cryptography to Hamiltonian mechanics, we command complicated data through a key data variable. We're human; we come to complicated issues through specific compelling interests. That data variable eventually drags us into deeper, less personal territory. Abel, Galois and Riemann knew that; though some versions of algebraic geometry from the late 1970s lost it. A choice data variable (function to the Riemann sphere) to extract information brings the tool of finite group theory. Giving such relations structure, and calling for advanced tools and interpretations, is the study of their moduli — the main goal of the papers of this volume. Applications in this volume are to analyzing properties of the absolute Galois group Gq (Part I) and to describing systems of relations over a finite field (Part II). Such applications have us looking at finite group theory and related algebra (Parts III and IV) in a new way. §1 reminds of classical problems that motivated those authors whose papers appear in this volume. We dramatize research events around the 1995 Seattle Con• ference, Recent Developments in the Inverse Galois Problem, in the Contemporary Math series of AMS, in §2. This also has reader aids for connecting that 1995 vol• ume to this volume's papers. In this one sees how group theory applies to modern applications. Conversely, when we see the effect of studying moduli of algebraic relations, it changes how we perceive group theory classification results. §3 points to applications addressed by particular papers in this volume. This is inadequate to explain either the motivations of the authors or the connections between the pa• pers. To get the results out to the public, the author of this prelude has added some necessary background to the Part I papers in §2. He will, however, reference this volume, in a later paper that ties the finite group theory, Lie theory and Tannakian viewpoints together. 2000 Mathematics Subject Classification. Primary 11F32, 11G18, 11R58; Secondary 20B05, 20C25, 20D25, 20E18, 20F34. Author support from MSRI, NSF #DMS-9970676 and a senior research Alexander von Hum• boldt award. © 2002 American Mathematical Society vii Vlll M. FRIED 1. Algebraic relations, moduli and GQ We use the word moduli to mean working smartly with gobs of algebraic rela• tions. As in our abstract we intend that the relations (algebraic curves) come with a data variable (function). An elliptic curve described by the equation y2 = x3+ax+b with x as its data variable is an appropriate example. So too is the u>-sphere with a choice of rational function f(w) — z. Applications demand understanding more than one relation at a time. We often must pick special equations from a mass of candidates. Typical is the quest for a good chunk of specific relations to have a chosen definition field (Q or a p-adic field). The j and A lines of classical complex variables inform the wise investigator of algebraic relations daily. Was there ever a greater joining of group theory to complex variables than when Galois applied his famous solvability criteria to modular curves [Ri96, last pages]? This extends that tradition where classical modular functions help us understand new relations. We closely relate to quotients of Siegel upper half space, significant moduli spaces. Taking advantage of data from a function on a curve immensely increases applica• tions. Further, we see from a different view problems that plague progress on Siegel modular functions. Relations with a data variable have more discrete invariants than the genus of a curve. Riemann called his favorite example half-canonical classes. These helped him pick the right theta function for his monumental solution to Jacobi's inversion problem. Serre connected Schur multipliers of alternating groups to half-canonical classes. His results hastened investigations of algebraic relations with groups having a nontrivial center. The most
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