The Riemann Hypothesis in Characteristic p in Historical Perspective

by Peter Roquette

Overview

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.

Product Details

ISBN-13:	9783319990668
Publisher:	Springer International Publishing
Publication date:	09/29/2018
Series:	Lecture Notes in Mathematics , #2222
Edition description:	1st ed. 2018
Pages:	235
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

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.

Table of Contents

- Overture.- Setting the stage.- The Beginning: Artin’s Thesis.- Building the Foundations.- Enter Hasse. - Diophantine Congruences. - Elliptic Function Fields. - More on Elliptic Fields. - Towards Higher Genus. - A Virtual Proof. - Intermission. - A.Weil. - Appendix. - References. - Index.

Editorial Reviews

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)

From the Publisher