© 2018

The Riemann Hypothesis in Characteristic p in Historical Perspective


Part of the Lecture Notes in Mathematics book series (LNM, volume 2222)

Also part of the History of Mathematics Subseries book sub series (HISTORYMS, volume 2222)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Peter Roquette
    Pages 1-8
  3. Peter Roquette
    Pages 9-12
  4. Peter Roquette
    Pages 13-37
  5. Peter Roquette
    Pages 39-54
  6. Peter Roquette
    Pages 55-57
  7. Peter Roquette
    Pages 59-79
  8. Peter Roquette
    Pages 81-106
  9. Peter Roquette
    Pages 107-129
  10. Peter Roquette
    Pages 131-156
  11. Peter Roquette
    Pages 157-179
  12. Peter Roquette
    Pages 181-202
  13. Peter Roquette
    Pages 203-213
  14. Peter Roquette
    Pages 215-219
  15. Back Matter
    Pages 221-235

About this book


This book tells the story of the Riemann hypothesis for function fields (or curves) starting with Artin's 1921 thesis, covering Hasse's work in the 1930s on elliptic fields and more, and concluding with Weil's final proof in 1948. The main sources are letters which were exchanged among the protagonists during that time, found in various archives, mostly the University Library in Göttingen. The aim is to show how the ideas formed, and how the proper notions and proofs were found, providing a particularly well-documented illustration of how mathematics develops in general. The book is written for mathematicians, but it does not require any special knowledge of particular mathematical fields.


Riemann hypothesis in characteristic p Emil Artin Helmut Hasse André Weil History of Number Theory in the 20th century

Authors and affiliations

  1. 1.Mathematical InstituteHeidelberg UniversityHeidelbergGermany

About the authors

Roquette studierte in Erlangen, Berlin und Hamburg und wurde 1951 an der Universität Hamburg bei Helmut Hasse promoviert, Ab 1967 ist er Professor an der Ruprecht-Karls-Universität Heidelberg, an der er 1996 emeritiert wurde. Roquette arbeitet über Zahl- und Funktionenkörper und speziell lokale p-adische Körper. Er wandte auch Methoden der Modelltheorie (Nonstandard Arithmetic) in der Zahlentheorie an, teilweise noch mit Abraham Robinson.. Er hat auch eine Reihe von Arbeiten zur Geschichte der Mathematik, insbesondere der Schulen von Helmut Hasse und Emmy Noether veröffentlicht. Roquette war 1975 Mitherausgeber der gesammelten Abhandlungen von Helmut Hasse und gab eine Zahlentheorie-Vorlesung von Erich Hecke aus dem Jahr 1920 neu heraus. Roquette ist seit 1978 Mitglied der Heidelberger Akademie der Wissenschaften[3] und seit 1985 der Deutschen Akademie der Naturforscher Leopoldina[4] sowie Ehrendoktor der Universität Duisburg-Essen und Ehrenmitglied der Mathematischen Gesellschaft Hamburg.

Bibliographic information

  • Book Title The Riemann Hypothesis in Characteristic p in Historical Perspective
  • Authors Peter Roquette
  • Series Title Lecture Notes in Mathematics
  • Series Abbreviated Title Lect.Notes Mathematics
  • DOI
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-99066-8
  • eBook ISBN 978-3-319-99067-5
  • Series ISSN 0075-8434
  • Series E-ISSN 1617-9692
  • Edition Number 1
  • Number of Pages IX, 235
  • Number of Illustrations 15 b/w illustrations, 0 illustrations in colour
  • Topics History of Mathematical Sciences
    Number Theory
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking


“The book will be read by mathematicians and historians of mathematics beyond those whose primary interests are in the fields discussed here, and one could only wish that more people knew enough mathematics to follow the history it considers.” (Arkady Plotnitsky, Isis, Vol. 111 (2), 2020)

“This is a rich and illuminating study of the mathematical developments over the period 1921-1942 that led to the proof by André Weil of the Riemann Hypothesis for algebraic function fields over a finite field of characteristic p (RHp). … Mathematicians with some knowledge of modern algebra and field theory will follow the main thread of the story, since the author avoids a heavily technical discussion.” (E. J. Barbeau, Mathematical Reviews, July, 2019)

“The book is very pleasant to read and should be consulted by any one interested in history, in function fields or in general in the RH in any characteristic. The book can be used by specialists and by non-specialists as a brief but very interesting introduction to function fields including its relation with algebraic geometry. … The summaries give a good abstract of the book.” (Gabriel D. Villa Salvador, zbMath 1414.11003, 2019)