GRADUATE STUDIES IN MATHEMATICS 177 Differential Galois Theory through Riemann-Hilbert Correspondence An Elementary Introduction Jacques Sauloy American Mathematical Society https://doi.org/10.1090//gsm/177 Differential Galois Theory through Riemann-Hilbert Correspondence An Elementary Introduction GRADUATE STUDIES IN MATHEMATICS 177 Differential Galois Theory through Riemann-Hilbert Correspondence An Elementary Introduction Jacques Sauloy American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staffilani Jeff A. Viaclovsky The photographs that appear on the dedication page are reprinted with permission. 2010 Mathematics Subject Classification. Primary 12H05, 30F99, 34-01, 34A30, 34Mxx. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-177 Library of Congress Cataloging-in-Publication Data Names: Sauloy, Jacques. Title: Differential Galois theory through Riemann-Hilbert correspondence : an elementary intro- duction / Jacques Sauloy. Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series: Gradu- ate studies in mathematics ; volume 177 | Includes bibliographical references and index. Identifiers: LCCN 2016033241 Subjects: LCSH: Galois theory. | Riemann-Hilbert problems. | AMS: Field theory and polynomi- als – Differential and difference algebra – Differential algebra. msc | Functions of a complex variable – Riemann surfaces – None of the above, but in this section. msc | Ordinary differ- ential equations – Instructional exposition (textbooks, tutorial papers, etc.). msc | Ordinary differential equations – General theory – Linear equations and systems, general. msc | Ordinary differential equations – Differential equations in the complex domain – Differential equations in the complex domain. msc Classification: LCC QA214 .S28 2016 | DDC 512/.32–dc23 LC record available at https://lccn.loc.gov/2016033241 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. 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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 http://www.ams.org/ 10987654321 212019181716 This book is dedicated to the Department of Mathematics of Wuda, in particular to its director Chen Hua, and to Wenyi Chen for giving me the occasion to teach this course and Jean-Pierre Ramis for giving me the ability to do so. The math buildings at Wuda (2012) and at Toulouse University (2013) Contents Foreword xiii Preface xv Introduction xvii Index of notation xix Part 1. A Quick Introduction to Complex Analytic Functions Chapter 1. The complex exponential function 3 §1.1. The series 3 §1.2. The function exp is C-derivable 4 §1.3. The exponential function as a covering map 7 §1.4. The exponential of a matrix 8 §1.5. Application to differential equations 10 Exercises 12 Chapter 2. Power series 15 §2.1. Formal power series 15 §2.2. Convergent power series 20 §2.3. The ring of power series 22 §2.4. C-derivability of power series 23 §2.5. Expansion of a power series at a point =0 25 §2.6. Power series with values in a linear space 26 Exercises 27 vii viii Contents Chapter 3. Analytic functions 29 §3.1. Analytic and holomorphic functions 29 §3.2. Singularities 32 §3.3. Cauchy theory 33 §3.4. Our first differential algebras 36 Exercises 37 Chapter 4. The complex logarithm 39 §4.1. Can one invert the complex exponential function? 39 §4.2. The complex logarithm via trigonometry 40 §4.3. The complex logarithm as an analytic function 41 §4.4. The logarithm of an invertible matrix 42 Exercises 44 Chapter 5. From the local to the global 45 §5.1. Analytic continuation 45 §5.2. Monodromy 47 §5.3. A first look at differential equations with a singularity 50 Exercises 52 Part 2. Complex Linear Differential Equations and their Monodromy Chapter 6. Two basic equations and their monodromy 57 §6.1. The “characters” zα 57 §6.2. A new look at the complex logarithm 70 §6.3. Back again to the first example 74 Exercises 75 Chapter 7. Linear complex analytic differential equations 77 §7.1. The Riemann sphere 77 §7.2. Equations of order n and systems of rank n 81 §7.3. The existence theorem of Cauchy 87 §7.4. The sheaf of solutions 89 §7.5. The monodromy representation 91 §7.6. Holomorphic and meromorphic equivalences of systems 95 Exercises 101 Chapter 8. A functorial point of view on analytic continuation: Local systems 103 Contents ix §8.1. The category of differential systems on Ω 103 §8.2. The category Ls of local systems on Ω 105 §8.3. A functor from differential systems to local systems 107 §8.4. From local systems to representations of the fundamental group 109 Exercises 113 Part 3. The Riemann-Hilbert Correspondence Chapter 9. Regular singular points and the local Riemann-Hilbert correspondence 117 §9.1. Introduction and motivation 118 §9.2. The condition of moderate growth in sectors 120 §9.3. Moderate growth condition for solutions of a system 123 §9.4. Resolution of systems of the first kind and monodromy of regular singular systems 124 §9.5. Moderate growth condition for solutions of an equation 128 §9.6. Resolution and monodromy of regular singular equations 132 Exercises 135 Chapter 10. Local Riemann-Hilbert correspondence as an equivalence of categories 137 §10.1. The category of singular regular differential systems at 0 138 §10.2. About equivalences and isomorphisms of categories 139 §10.3. Equivalence with the category of representations of the local fundamental group 141 §10.4. Matricial representation 142 Exercises 144 Chapter 11. Hypergeometric series and equations 145 §11.1. Fuchsian equations and systems 145 §11.2. The hypergeometric series 149 §11.3. The hypergeometric equation 150 §11.4. Global monodromy according to Riemann 153 §11.5. Global monodromy using Barnes’ connection formulas 157 Exercises 159 Chapter 12. The global Riemann-Hilbert correspondence 161 §12.1. The correspondence 161 x Contents §12.2. The twenty-first problem of Hilbert 162 Exercises 166 Part 4. Differential Galois Theory Chapter 13. Local differential Galois theory 169 §13.1. The differential algebra generated by the solutions 170 §13.2. The differential Galois group 172 §13.3. The Galois group as a linear algebraic group 175 Exercises 179 Chapter 14. The local Schlesinger density theorem 181 §14.1. Calculation of the differential Galois group in the semi-simple case 182 §14.2. Calculation of the differential Galois group in the general case 186 §14.3. The density theorem of Schlesinger in the local setting 188 §14.4. Why is Schlesinger’s theorem called a “density theorem”? 191 Exercises 192 Chapter 15. The universal (fuchsian local) Galois group 193 §15.1. Some algebra, with replicas 194 §15.2. Algebraic groups and replicas of matrices 196 §15.3. The universal group 199 Exercises 200 Chapter 16. The universal group as proalgebraic hull of the fundamental group 201 §16.1. Functoriality of the representationρ ˆA ofπ ˆ1 201 §16.2. Essential image of this functor 203 §16.3. The structure of the semi-simple component ofπ ˆ1 207 §16.4. Rational representations ofπ ˆ1 213 §16.5. Galois correspondence and the proalgebraic hull ofπ ˆ1 214 Exercises 216 Chapter 17. Beyond local fuchsian differential Galois theory 219 §17.1. The global Schlesinger density theorem 220 §17.2. Irregular equations and the Stokes phenomenon 221 §17.3. The inverse problem in differential Galois theory 226 Contents xi §17.4. Galois theory of nonlinear differential equations 227 Appendix A. Another proof of the surjectivity of exp : Matn(C) → GLn(C) 229 Appendix B. Another construction of the logarithm of a matrix 233 Appendix C. Jordan decomposition in a linear algebraic group 237 §C.1. Dunford-Jordan decomposition of matrices 237 §C.2. Jordan decomposition in an algebraic group 241 Appendix D. Tannaka duality without schemes 243 §D.1. One weak form of Tannaka duality 245 §D.2. The strongest form of Tannaka duality 246 §D.3. The proalgebraic hull of Z 248 §D.4. How to use tannakian duality in differential Galois theory 251 Appendix E. Duality for diagonalizable algebraic groups 255 §E.1. Rational functions and characters 255 §E.2. Diagonalizable groups and duality 257 Appendix F. Revision problems 259 §F.1. 2012 exam (Wuhan) 259 §F.2. 2013 exam (Toulouse) 260 §F.3. Some more revision problems 263 Bibliography 267 Index 271 Foreword Nowadays differential Galois theory is a topic appearing more and more in graduate courses. There are several reasons. It mixes fundamental objects from very different areas of mathematics and uses several interesting the- ories. Moreover, during the last decades, mathematicians have discovered a lot of important and powerful applications of differential Galois theory. Some are quite surprising, such as some criteria for the classical problem of integrability of dynamical systems in mechanics and physics.