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2018 | Buch

Overture Kapitel lesen Erstes Kapitel lesen

The Riemann Hypothesis in Characteristic p in Historical Perspective

verfasst von: Prof. Peter Roquette

Verlag: Springer International Publishing

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Overture
Abstract
Mathematics is, on the one hand, a cumulative science. Once a mathematical theorem has been proved to be true then it remains true forever: it is added to the stock of mathematical discoveries which has piled up through the centuries and it can be used to proceed still further in our pursuit of knowledge.
Peter Roquette
Chapter 2. Setting the Stage
Abstract
I have written this book for mathematicians but I do not assume that the reader is familiar with the basic notions and terminology of the theory of algebraic function fields in the 1930s. The present section provides an introduction to this. At the same time I would like to fix the notations which I will be using in this book.
Peter Roquette
Chapter 3. The Beginning: Artin’s Thesis
Abstract
Emil Artin (1898–1962) was born in Vienna. He was brought up in Reichenberg, a German speaking town in Northern Bohemia belonging to the Austro-Hungarian empire. (The town is now called Liberec, in the Czech Republic). In 1916 he enrolled at the University of Vienna where, among others, he attended a lecture course by Ph. Furtwängler. After one semester of study he was drafted to the army. In January 1919 he entered the University of Leipzig. I have taken this information from Artin’s own hand-written vita that he submitted together with his thesis to the Faculty at Leipzig University. There in June 1921 he obtained his Ph.D. with Herglotz as his thesis advisor.
Peter Roquette
Chapter 4. Building the Foundations
Abstract
Friedrich Karl Schmidt (1901–1976) studied at the University of Freiburg where he did his Ph.D. examination in May of 1925.
Peter Roquette
Chapter 5. Enter Hasse
Abstract
Helmut Hasse (1898–1979) was of the same age as Artin.
Peter Roquette
Chapter 6. Diophantine Congruences
Abstract
Harold Davenport had been introduced to Hasse in 1930 by Louis J. Mordell. This came about as follows:
Peter Roquette
Chapter 7. Elliptic Function Fields
Abstract
We all know that a good way to study a mathematical subject is to give a lecture course about it. The necessity to arrange the theory in a systematic way and to explain to the audience the various connections between the different results, often leads to new insights and, in consequence, to new results.
Peter Roquette
Chapter 8. More on Elliptic Fields
Abstract
While Hasse worked on the RHp for elliptic fields he and his team discovered several properties of elliptic fields in characteristic p which were new and important although they were not absolutely necessary for the proof of RHp. In this chapter I am going to discuss some of them.
Peter Roquette
Chapter 9. Towards Higher Genus
Abstract
As I have told in Sect. 6.​2 Hasse originally was interested in the estimate of solutions of diophantine congruences. It was Artin who in November of 1932, when Hasse was visiting Hamburg, told him that the estimating problem was pointing to the RHp. It appears that at that time Hasse did not yet believe in the general validity of the RHp for all function fields with higher genus g > 1. (See page 65.)
Peter Roquette
Chapter 10. A Virtual Proof
Abstract
In this chapter I would like to interrupt the historic line in order to put into evidence what I just said, namely that the proof of RHp could have been found already in 1937, in the framework of the theory of function fields. I will present here such a proof. In principle it can be regarded as a translation of Severi’s proof from the language of algebraic geometry into the language of algebra. But I will not use any knowledge of the terminology and results of algebraic geometry. I shall use those notions and facts from the theory of function fields which were available to and preferred by Hasse at the time of the Göttingen workshop which I have discussed above.
Peter Roquette
Chapter 11. Intermission
Abstract
If one would compare our story with a concert, then Artin’s thesis together with F.K. Schmidt’s paper would pass as the first and second part of the Introduction (Chaps. 3 and 4). Hasse’s work on the elliptic case (Chap. 7) would be the first movement allegro assai with the theme set by Davenport (Chap. 6). Deuring’s theory of correspondences (Chap. 9) would pass as the second movement sostenuto, covering the attempt towards higher genus. The insertion of the virtual proof (Chap. 10) may go as scherzo allegretto.
Peter Roquette
Chapter 12. A. Weil
Abstract
André Weil (1906–1998) was 8 years younger than Hasse. He was born and raised in Paris. He received his doctorate 1928 at the University of Paris, supervised by Hadamard, with his thesis “Arithmetic of algebraic curves” where he proved his part of what today is called the Mordell-Weil Theorem. His name appeared already several times in our story since he had exchanged letters with Hasse and had early shown interest in the RHp.
Peter Roquette
Chapter 13. Appendix
Abstract
With the appearance of Weil’s above mentioned three books, the RHp was settled and our story comes to an end. But the mathematical development inspired by this or that item of our story persists and is still present. From the numerous literature in this direction I will mention here three papers only:
Peter Roquette
Correction to: The Riemann Hypothesis in Characteristic p in Historical Perspective
Peter Roquette
Backmatter
Metadaten
Titel
The Riemann Hypothesis in Characteristic p in Historical Perspective
verfasst von
Prof. Peter Roquette
Copyright-Jahr
2018
Electronic ISBN
978-3-319-99067-5
Print ISBN
978-3-319-99066-8
DOI
https://doi.org/10.1007/978-3-319-99067-5

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