Manuscripts by Peter Roquette

Comments and corrections are welcomed.

current      2009      2005     2002     1999    

May 10, 2021, An introduction to Deuring's theory of constant reductions .

This article is contained in the volume 767 (2021) of "Contemporary Mathematics"  which is dedicated to Gerhard Frey.

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August 15, 2017, Riemann hypothesis in characteristic p in historical perspective.

This is a preview of the introduction of a book which  appeared in Springer Verlag.

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September 4, 2014  Jacques Herbrand and his Lemma.

This article introduces the reader to the group theoretical Lemma of Herbrand, its origin and its use in class field theory. At the same time it traces Herbrand's acitivities during his stay in Germany 1930/31. Published in Mitteilungen der Mathematischen Gesellschaft in Hamburg vol.34 (2014).

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March 8, 2014  Herbrand's Lemma as a Hasse Diagram .

Recently, among the Hasse papers which are kept at the University Library in Göttingen, I found a sheet of paper carrying Hasse's remark  "Roquette, December 1949". It appears that the paper contained a presentation of Herbrand's Lemma which I as a student had given that year in Hasse's Number Theory seminar at the Humboldt University in Berlin . Hasse seems to have liked it since he had kept it in his files through all these years. Perhaps it may be of general interest to see that the proof of Herbrand's Lemma, which nowadays is usually embedded into algebraic cohomology, can be given just in form of a diagram, understandable without text at least by those who are familiar with Hasse diagrams. The original paper is handwritten and contains the diagrams only. I have now transcribed it with TeX and added some comments how to read those diagrams. Addendum (September 9, 2014): Same for the generalized Herbrand Lemma. (3pp.)  

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Sep 25, 2013  Contemplations of an octogenarian mathematician .

This is the text of a talk, read at the Heidelberg Mathematical Institute on September 25, 2013, on the occasion of a visit of 20 young mathematicians from all over the world who attended the Heidelberg Laureate Forum.  (14pp.)  

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July 30, 2013  Drei Frauenschicksale in den Mathematischen Wissenschaften.

Es handelt sich um die Manuskripte von drei Vorträgen vor einem offenen Hörerkreis (nicht vor Mathematikern). Inhaltlich enthalten diese Vorträge nichts Neues, was nicht schon in der einschlägigen Literatur zu finden ist.

1. Emmy Noether (1882 - 1935), 25 pp.         pdf

2. Sonja Kowalevskaja (1850 - 1891), 32 pp.      pdf

3. Sophie Germain (1776-1831),   in preparation.            back to list of contents

January 24, 2013 Contributions to the History of Number Theory in the 20th Century    (Book, 278 pp.)

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August 12, 2012 Helmut Hasse und die Familie Mendelssohn (version. of November 19, 2012).

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Abstract: Vor einer Reihe von Jahren erhielt ich von Martin Kneser, dem Schwiegersohn von Helmut Hasse, eine Mappe mit einer Reihe von Papieren aus dem Hasse-Nachlass, die mehr oder minder privaten Charakter tragen und wohl deshalb von Hasse nicht zur Übergabe an das Archiv der Göttinger Universität vorgesehen waren. Unter diesen fand sich ein Blatt mit einem sogenannten Familienstammbaum der Familie Mendelssohn. Mitteilungen der Mathematischen Gesellschaft in Hamburg Band XXXIII (2013) 197-200.

June 18, 2012 The Riemann hypothesis in characteristic p, its origin and development. Part 4: Davenport-Hasse fields.  (Mitteilungen der Mathematischen Gesellschaft in Hamburg, Band 32 (2012) 145-210 .)

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Abstract: This is the fourth part of a larger work on the history of the so-called Riemann hypothesis in characteristic p. We report on the work of Davenport and Hasse on zeta functions of the so-called Davenport-Hasse function fields. Their zeros can be obtained by means of Gaussian sums. Thus the Riemann hypothesis was proved for a wide class of function fields of higher genus in the year 1934 already.We include a discussion of several other papers in the 1930s which came up in the context of this work. Besides Davenport and Hasse we will meet the names of Stickelberger, Artin, Witt, H.L.Schmid, Teichmüller, A.Weil, Chevalley and others.

May 31, 2012   Vortrag über Emmy Noether.

On May 27, 2010  I had given a lecture on Emmy Noether at the Mathematics Department of Heidelberg University. The lecture was directed to all students of the department, independent of their year of study. No special prerequisites were required. Originally I had planned to write up some kind of lecture notes to be handed out to the participants. However, other projects have prevented me to do so. Since I do not know when I will be able to complete those lecture notes, I have now decided to publish at least the slides (in German) which were shown during the lecture. Although the spoken text connecting these slides is missing, they carry also some information on their own and thus may be of some use. If you do have some questions it will be a pleasure to me to reply if possible. .    

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Jan 10, 2012   Heinrich-Wolfgang Leopoldt 1927-2011.

Obituary. (German)  für das Jahrbuch der Heidelberger Akademie der Wissenschaften.

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English version for  Journal of Number Theory:

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June 17, 2011   Cahit Arf and his invariant.

This article is the result of unifying and re-editing of two previous papers. For the convenience of the reader we have included an appendix containing simple proofs of Arf's main results in corrected form, pointing out that all the methods and facts which we use in these proofs can be found in Arf's paper already. Joint work with Falko Lorenz. The article has appeared in "Mitteilungen der Mathematischen Gesellschaft in Hamburg", vol. XXX (2011), 87-126.

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September 10, 2010   In Memoriam Ernst Steinitz.

In 1910, in volume 137 of Crelle's Journal there appeared a paper with the title "Algebraische Theorie der Körper" (Algebraic Theory of Fields). The author was Ernst Steinitz. We  use the occasion of a centenary to recall the impact which Steinitz's paper had upon the mathematicians of the time, and its role in the development of today's algebra. The article  has appeared in Crelle's Journal, vol. 2010, Ausg. 648, 1-10.

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March 19, 2010 Betr. Fotos Emmy Noether.

Ziel dieses Artikels ist es, darauf hinzuweisen, dass ein Foto, das in vergrößerter, kolorierter und bearbeiteter Form kürzlich auf einem Buchtitel über Emmy Noether aufgetaucht ist,  nicht Emmy Noether darstellt.

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English translation: June 21, 2010 About Photos of Emmy Noether.

The aim of this article is to point out that a certain photo, which recently has appeared in enlarged, colored and  modified form on the cover of a book on Emmy Noether, does not show Emmy Noether.

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June 13, 2008 Emil Artin - Helmut Hasse. Their correspondence 1923 - 1934. (Version of Nov 10, 2008)

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Abstract: This book reproduces the complete extant correspondence between Emil Artin and Helmut Hasse. There are 49 such letters. The correspondence reflects a crucial period in the development of 20th century algebra and number theory, in particular class field theory and Reciprocity Laws. Details of proofs appear alongside with conjectures and speculations. Also discussed are questions of textbook presentation, e.g., for quadratic forms and for units. The editors have amended each letter by a detailed commentary. The Artin-Hasse correspondence is a rich source for those who are interested in the rise and the development of mathematical notions and ideas. This book has appeared in the Universitätsverlag Göttingen as a companion to the book on the correspondence between Helmut Hasse and Emmy Noether which had appeared in 2006. For the latter see Nr. 35.
 

June 10, 2008 Numbers and models, standard and nonstandard.  

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This had been the manuscript for a talk at the ``Algebra Days'', May, 2008, in Antalya. I talked about my personal recollections of Abraham Robinson. The manuscript had been temporarily withdrawn from my homepage. This is the extended and corrected version of Apr 28, 2010, and it has appeared in the Journal Mathematische Semesterberichte 57 (2010) 185-199.

March 27, 2008 Über Wolfgang Wichmann.  

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Wolfgang Wichmann war einer der letzten Schüler von Emmy Noether in Göttingen. In seiner Dissertation (1934) findet sich ein einfacher Beweis für die Funktionalgleichung der Hey-schen Zetafunktion einer Divisionsalgebra. Auf diesen Beweis, der offenbar in Vergessenheit geraten war, hat kürzlich Falko Lorenz aufmerksam gemacht (Mitt. der Math. Ges. in Hamburg, Band XXVII, 2008). Unser Artikel gibt als Nachtrag zur Arbeit von Lorenz einige persönliche Daten über Wichmann, soweit wir sie heute noch feststellen konnten. (4 Seiten.)

January 22, 2008 Emmy Noether: Die Gutachten.  

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This is a documentation of the testimonials (Gutachten) which were written by prominent mathematicians for Emmy Noether in the critical years 1933/34. (Modified version of 10 February 2008.) The Einleitung (introduction) is published in: Mitteilungen der Deutschen Mathematiker Vereinigung 16 (2008) 118-123.

May 15, 2007 Emmy Noether and Hermann Weyl.  

Sep 18, 2006 The Riemann hypothesis in characteristic p, its origin and development. Part 3: The elliptic case.  Mitteilungen der Mathematischen Gesellschaft in Hamburg 25 (2006) 103-176

May 12, 2006    Helmut Hasse - Emmy Noether. Their correspondence 1925 - 1935. Edited and commented by Franz Lemmermeyer and Peter Roquette.  Published  by "Universitätsverlag Göttingen" 2006. 

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Abstract: This book reproduces the complete extant correspondence between Emmy Noether and Helmut Hasse. There are 82 such letters, of which 79 are from Noether to Hasse, dating from 1925 until Noether‘s sudden death in 1935. The correspondence reflects a crucial period in the development of 20th century algebra and number theory, in particular class field theory. Details of proofs appear alongside with conjectures and speculations. Also discussed are questions of textbook presentation, e.g., of Galois theory. Aside from mathematical details, the spontaneity of Noether‘s style allows many glimpses at the image that Emmy Noether and Helmut Hasse had of the topics they were working in. The Hasse – Noether correspondence is a rich source for those who are interested in the rise and the development of mathematical notions and ideas. Each letter is accompanied by a detailed commentary supplied by the editors. For the convenience of the reader, numerous cross-references, extended indexes, and short biographies of all persons mentioned in the correspondence have been added.

Mar 12, 2006    Zu Emmy Noethers Geburtstag. Einige neue Noetheriana.    Mitteilungen der Deutschen Mathematiker-Vereinigung 15/1 (2007) 15-21

The original manuscript of March 12, 2006 has been replaced by a new, extended version. (16 pages.)

(Remark: Recently I have found a number of other, hitherto unknown, documents on the life and work of Emmy Noether. These will be published elsewhere, and will be available on this website eventually.)

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Feb 25, 2006    Helmut Schaefer 1925-2005.  Nachruf im Jahrbuch 2005 der Heidelberger Akademie der Wissenschaften, gemeinsam mit Manfred Wolff.  

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Mar 09, 2005    From FLT to finite groups. The remarkable career of Otto Grün. Jahresbericht der Deutschen Mathematiker-Vereinigung  107 (2005) 117-154

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Abstract: Every student who starts to learn group theory will soon be confronted with the theorems of Grün. Immediately after their publication in the mid 1930s these theorems found their way into group theory textbooks, with the comment that those theorems are of fundamental importance in connection with the classical Sylow theorems. But little is known about the mathematician whose name is connected with those theorems.   We shall report about the remarkable mathematical career of Otto Grün who, as an amateur mathematician without having had the opportunity to attend university, published his first paper (out of 26) when he was 46.

Jan 21, 2005    Laudatio für B. H. Matzat

Manuskript einer Rede, gehalten am 21.Januar 2005 in einem Mathematischen Festkolloquium in Heidelberg, aus Anlass des 60. Geburtstages von B. Heinrich Matzat. 12 Seiten.

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Each year there is scheduled a special colloquium lecture at the Middle East Technical University in Ankara, commemorating Cahit Arf who had served as a professor for many years at METU. In the year 2004 the invited speaker was Robert Langlands, and I had been asked to chair his lecture. The text of this manuscript was read to introduce the speaker.

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Abstract: This article is based on letters and documents found at the University library Göttingen and other archives, centered around the main structure theorem in the theory of algebras, including a chapter on the contributions of Albert to this problem. It covers mainly the years around 1931. This is the extended and revised manuscript of our preliminary version of   March 17, 2001 .  (Some misprints in the version of Feb 28, 2004 have been corrected on Dec 23, 2004.)

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Abstract: The origin of this note was the attempt to answer a question of Moshe Jarden who had asked me:

''Hat jeder Zahlkörper K ein endliches Einbettungsproblem für Gal(K), das lokal lösbar aber global nicht lösbar ist?''

As a first reaction I referred him to an old paper of Hoechsmann on the embedding problem. But after a second reading of Hoechsmann's paper I found that the answer to Moshe's question - which is affirmative - is not explicitly stated there. Although the answer can be readily derived using Hoechsmann's ideas, it is perhaps not without interest to do this explicitly. This is what I propose in this note.

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Abstract: This paper is to be regarded as the first part of a larger project which will give a comprehensive view of the history around the Riemann hypothesis for function fields. (For a preliminary version see the manuscript of   Dec 15, 1997 .) This Part 1 is dealing with the development before Hasse's contributions to the Riemann hypothesis. We are trying to explain what he could build upon. The time interval covered will be roughly between 1921 and 1931. We start with Artin's thesis of 1921 where the Riemann hypothesis for function fields was spelled out and discussed for the first time, namely in the case of quadratic function fields. We will describe the activities following Artin's thesis until F.K.Schmidt's classical paper 1931 on the Riemann-Roch theorem and the zeta function of an arbitrary function field. Finally we will review Hasse's paper in 1934 where he gives a summary about all what was known at that time about zeta functions of function fields.

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Abstract: The legacy of Helmut Hasse, consisting of letters, manuscripts and other papers, is kept at the University Library at Göttingen. Hasse had an extensive correspondence; he liked to exchange mathematical ideas, results and methods freely with his colleagues. There are more than 8000 letters preserved. Although not all of them are of equal mathematical interest, searching through this treasure can help us to assess the development of Number Theory through the 1920's and 1930's. Unfortunately, most of the correspondence is preserved on one side only, i.e., the letters sent to Hasse are available whereas many of the letters which had been sent from him, often handwritten, seem to be lost. So we have to interpolate, as far as possible, from the replies to Hasse and from other context, in order to find out what he had written in his outgoing letters. In the present article I shall report about the letters which document the activity around the Brauer-Hasse-Noether Theorem in Number Theory; this covers the years around 1931. Besides of the letters from the Hasse legacy, I shall also use some letters from Emmy Noether to Richard Brauer which are preserved at the Bryn Mawr College Library (Pennsylvania, USA).