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Arithmetic Geometry Over Global Function Fields Advanced Courses in Mathematics CRM Barcelona Gebhard Böckle David Burns David Goss Dinesh Thakur Fabien Trihan Douglas Ulmer Arithmetic Geometry over Global Function Fields Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica Managing Editor: Carles Casacuberta More information about this series at http://www.springer.com/series/5038 Gebhard Böckle • David Burns • David Goss Dinesh Thakur • Fabien Trihan • Douglas Ulmer Arithmetic Geometry over Global Function Fields Editors for this volume: Francesc Bars (Universitat Autònoma de Barcelona) Ignazio Longhi (Xi’an Jiaotong-Liverpool University) Fabien Trihan (Sophia University, Tokyo) Gebhard Böckle David Burns Interdisciplinary Center for Scientific Computing Department of Mathematics Universität Heidelberg King’s College London Heidelberg, Germany London, UK David Goss Dinesh Thakur Department of Mathematics Department of Mathematics The Ohio State University University of Rochester Columbus, OH, USA Rochester, NY, USA Fabien Trihan Douglas Ulmer Department of Information School of Mathematics and Communication Sciences Georgia Institute of Technology Sophia University Atlanta, GA, USA Tokyo, Japan ISSN 2297-0304 ISSN 2297-0312 (electronic) ISBN 978-3-0348-0852-1 ISBN 978-3-0348-0853-8 (eBook) DOI 10.1007/978-3-0348-0853-8 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2014955449 Mathematics Subject Classification (2010): Primary: 11R58; Secondary: 11B65, 11G05, 11G09, 11G10, 11G40, 11J93, 11R23, 11R65, 11R70, 14F05, 14F43, 14G10, 33E50 © Springer Basel 2014 This work is subject to copyright. 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Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com) Contents Foreword ..................................... xiii Cohomological Theory of Crystals over Function Fields and Applications Gebhard B¨ockle Introduction.................................. 3 Notation.................................... 7 1 First Basic Objects .............................. 9 1.1 τ-sheaves............................... 9 1.2 (Algebraic) Drinfeld A-modules................... 11 1.3 A-motives............................... 12 2 A-crystals ................................... 15 2.1 MotivationII............................. 15 2.2 Localization.............................. 18 2.3 Localizationforabeliancategories.................. 20 2.4 Nilpotence............................... 21 2.5 A-crystals............................... 23 3 Functors on τ -sheaves and A-crystals ................... 25 3.1 Inverseimage............................. 25 3.2 Further functors deduced from functors on quasi-coherentsheaves........................ 28 3.3 Extensionbyzero........................... 29 4 Derived Categories and Derived Functors ................. 33 5 Flatness .................................... 37 5.1 Basicsonflatness........................... 37 5.2 Flatness under functors . 38 5.3 Representability of flat crystals . 39 v vi Contents 6TheL-function ................................ 41 6.1 Naive L-functions........................... 41 6.2 Crystalline L-functions........................ 42 6.3 Trace formulas for L-functions.................... 44 7 Proof of Anderson’s Trace Formula and a Cohomological Interpretation ........................ 47 7.1 TheCartieroperator......................... 47 7.2 Cartiersheaves............................ 48 7.3 Operatorsoftraceclass....................... 50 7.4 Anderson’straceformula....................... 51 7.5 ProofofTheorem6.13........................ 53 7.6 The crystalline trace formula for general (good) rings A ..... 54 8 Global L-functions for A-motives ...................... 57 8.1 Exponentiationofideals....................... 58 8.2 Definition and basic properties of the global L-function...... 60 8.3 Global L-functionsatnegativeintegers............... 62 8.4 Meromorphyandentireness..................... 63 8.5 The global Carlitz–Goss L-functionoftheaffineline....... 64 9 Relation to Etale´ Sheaves .......................... 73 9.1 Anequivalenceofcategories..................... 73 9.2 AresultofGossandSinnott..................... 77 10 Drinfeld Modular Forms ........................... 85 10.1 AmodulispaceforDrinfeldmodules................ 86 10.2 Anexplicitexample......................... 87 10.3 Drinfeldmodularformsviacohomology.............. 91 10.4 Galois representations associated to Drinfeld modular forms . 96 10.5 Ramification of Galois representations associated to Drinfeld modularforms............................ 98 10.6 Drinfeld modular forms and Hecke characters . 100 10.7 Anextendedexample......................... 102 Appendix: Further Results on Drinfeld Modules ............... 108 A.1 Drinfeld A-modules over C∞ .................... 108 A.2 Torsion points and isogenies of Drinfeld modules . 110 A.3 Drinfeld–Hayesmodules....................... 111 A.4 TorsionpointsofDrinfeld–Hayesmodules............. 114 Bibliography ................................... 116 Contents vii On Geometric Iwasawa Theory and Special Values of Zeta Functions David Burns and Fabien Trihan with an appendix by Francesc Bars Introduction ................................... 121 1 Preliminaries ................................. 123 1.1 Relative algebraic K-theory and Iwasawa algebras . 123 1.2 Pro-coverings, pro-sheaves and perfect complexes . 126 2 Higher direct images ............................. 126 3 Proof of Theorem 2.1 ............................ 129 3.1 Finitegeneration........................... 129 3.2 TheHochschild–Serreexacttriangle................ 129 3.3 Animportantreductionstep..................... 131 3.4 Theabeliancase........................... 131 3.5 StrategyofBurnsandKato..................... 133 3.6 ApplicationtoTheorem2.1..................... 134 3.7 ProofofTheorem3.7......................... 136 4 Semistable abelian varieties over unramified extensions ......... 137 4.1 Hypothesesandnotations...................... 137 4.2 Statementofthemainresults.................... 138 5 Proof of Theorem 4.1 ............................ 141 5.1 The complexes N0 and S0 ...................... 141 5.2 Extending to the complexes S∞ and N∞ .............. 143 5.3 The complex Nar ........................... 144 5.4 The complex N∞ ........................... 145 5.5 Main Conjecture for A over K∞ ................... 146 6 Constant ordinary abelian varieties over abelian extensions ....... 147 6.1 The p-adic L-function of A/L .................... 148 6.2 Theinterpolationformula...................... 149 6.3 Main Conjecture for A over L .................... 150 7 Proof of Theorem 6.3 ............................ 151 7.1 Frobenius-Verschiebung decomposition . 151 7.2 Selmermodulesandclassgroups.................. 152 7.3 Functionalequations......................... 152 7.4 Completionoftheproof....................... 155 viii Contents 8 Explicit consequences ............................ 156 8.1 Weil-´etalecohomology........................ 156 8.2 Leadingtermformulasforaffinecurves............... 158 9 Proofs of Theorem 8.1 and Corollary 8.2 .................. 161 10 Fitting invariants and annihilation results ................. 165 Appendix: On Non-Noetherian Iwasawa Theory A.1 Thegeneralsetting.......................... 170 A.2 Divisorclassgroups.......................... 171 A.2.1Generalobservations..................... 171 A.2.2Pro-characteristicideals................... 172 A.2.3Themainconjecture..................... 172 A.3 Selmer groups in the p-cyclotomicextension............ 173 A.3.1Pro-Fittingideals....................... 173 A.3.2Pro-characteristicideals................... 174 Bibliography ................................... 177 The Ongoing Binomial Revolution David Goss Introduction ................................... 185 1 Early history ................................. 185 2 Newton, Euler, Abel, and
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