The Riemann Hypothesis in Characteristic p in Historical Perspective Peter Roquette Version of 3.7.2018 2 Preface This book is the result of many years' work. I am telling the story of the Riemann hypothesis for function fields, or curves, of characteristic p starting with Artin's thesis in the year 1921, covering Hasse's work in the 1930s on elliptic fields and more, until Weil's final proof in 1948. The main sources are letters which were exchanged among the protagonists during that time which I found in various archives, mostly in the University Library in G¨ottingen but also at other places in the world. I am trying to show how the ideas formed, and how the proper notions and proofs were found. This is a good illustration, fortunately well documented, of how mathematics develops in general. Some of the chapters have already been pre-published in the \Mitteilungen der Mathematischen Gesellschaft in Hamburg". But before including them into this book they have been thoroughly reworked, extended and polished. I have written this book for mathematicians but essentially it does not require any special knowledge of particular mathematical fields. I have tried to explain whatever is needed in the particular situation, even if this may seem to be superfluous to the specialist. Every chapter is written such that it can be read independently of the other chapters. Sometimes this entails a repetition of information which has al- ready been given in another chapter. The chapters are accompanied by \summaries". Perhaps it may be expedient first to look at these summaries in order to get an overview of what is going on. The preparation of this book had been supported in part by the Deutsche Forschungsgemeinschaft and the Marga und Kurt M¨ollgaard Stiftung.A number of colleagues and friends have shown interest and helped me with 3 4 their critical comments; it is impossible for me to mention all of them here. Nevertheless I would like to express my special thanks to Sigrid B¨oge,Karin Reich, Franz-Viktor Kuhlmann and Franz Lemmermeyer. References: Most of the letters and documents cited in this book are contained in the Handschriftenabteilung of the University Library of G¨ottin- gen (except if another source is mentioned), most but not all of them in the Hasse-files. The letters from Hasse to Davenport are contained in the archive of Trinity College, Cambridge. The letters from Hasse to Mordell are con- tained in the archive of King's College, Cambridge. The letters from Hasse to Fraenkel are in the archive of the Hebrew University, Jerusalem. I have translated the cited text into English for easier reading. Transcriptions of the full letters (and more) in their original language are available at my homepage: https://www.mathi.uni-heidelberg.de/∼roquette/ Contents 1 Overture 7 1.1 Why history of mathematics ? . 7 1.2 Artin and the intervention by Hilbert . 9 1.3 Hasse's project . 11 1.4 Weil's contribution . 14 Summary . 15 2 Setting the stage 17 2.1 Our Terminology . 17 2.2 The Theorem: RHp . 20 3 The beginning: Artin's thesis 23 3.1 Quadratic function fields . 24 3.1.1 The arithmetic part . 29 3.1.2 The analytic part . 33 3.2 Artin's letters to Herglotz . 39 3.2.1 Extension of the base field . 40 3.2.2 Complex multiplication . 41 3.2.3 Birational transformation . 45 3.3 Hilbert and the consequences . 46 5 CONTENTS 6 Summary . 47 3.4 Side remark: Gauss' last entry . 47 Summary . 54 4 Building the foundations 55 4.1 F.K.Schmidt . 55 4.2 Zeta function and Riemann-Roch Theorem . 60 4.3 F.K.Schmidt's L-polynomial . 65 4.3.1 Some comments . 66 4.4 The functional equation . 68 4.5 Consequences . 69 Summary . 73 5 Enter Hasse 75 6 Diophantine congruences 79 6.1 Davenport . 79 6.2 The challenge . 82 6.3 The Davenport-Hasse paper . 89 6.3.1 Davenport's letter and generalized Fermat fields . 90 6.3.2 Gauss sums . 94 6.3.3 Davenport-Hasse fields . 97 6.3.4 Stickelberger's Theorem . 99 6.4 Exponential sums . 100 Summary . 103 7 Elliptic function fields 105 7.1 The breakthrough . 105 CONTENTS 7 7.2 The problem . 108 7.3 Hasse's first proof: complex multiplication . 110 7.4 The second proof . 117 7.4.1 Meromorphisms and the Jacobian . 120 7.4.2 The double field . 124 7.4.3 Norm Addition formula . 126 7.4.4 The Frobenius operator . 132 7.5 Some Comments . 135 7.5.1 Rosati's anti-automorphism . 135 Summary . 136 8 More on elliptic fields 139 8.1 The Hasse invariant A . 139 8.2 Unramified cyclic extensions of degree p . 141 8.2.1 The Hasse-Witt matrix . 144 8.3 Group structure of the Jacobian . 145 8.3.1 Higher Derivations . 149 8.4 The structure of the endomorphism ring . 152 8.4.1 The supersingular case . 153 8.4.2 Singular invariants . 157 8.4.3 Elliptic subfields . 159 8.4.4 Good reduction . 160 8.5 Class field theory and complex multiplication . 164 Summary . 166 9 Towards higher genus 169 9.1 Preliminaries . 169 CONTENTS 8 9.2 The years 1934-35 . 173 9.2.1 Deuring . 176 9.2.2 More political problems . 179 9.3 Deuring's letter to Hasse . 182 9.3.1 Correspondences . 184 9.4 Hasse's letter to Weil . 189 9.5 Weil's reply and Lefschetz's note . 192 9.6 The workshop on algebraic geometry . 196 Summary . 202 10 A Virtual Proof 203 10.1 The quadratic form . 204 10.1.1 The double field . 204 10.1.2 The different . 206 10.2 Positivity . 214 10.2.1 Applying the Riemann-Hurwitz formula . 214 10.2.2 The discriminant estimate . 217 10.3 The RHp . 223 10.4 Some Comments . 225 10.4.1 Frobenius meromorphism. 225 10.4.2 Differents. 226 10.4.3 Integer differentials . 227 10.5 The geometric language . 227 Summary . 230 11 Intermission 231 11.1 Artin leaves . 232 CONTENTS 9 11.2 The Italian connection . 234 11.2.1 Severi . 234 11.2.2 The Volta congress . 238 11.3 The French Connection . 243 11.3.1 On function fields . 243 11.3.2 The book . 250 11.3.3 Paris and Strassbourg . 254 Summary . 259 12 A.Weil 261 12.1 Bonne Nouvelle . 262 12.2 The first note 1940 . ..