65  Manuscripts by Peter Roquette

(supported in part by the DFG and the Möllgaard-Stiftung)

for downloading as pdf-file  or dvi-file   (Last modified:  March 8, 2014)        to homepage

Comments and corrections are welcomed.


Contents:

current      2009      2005     2002     1999    

60 March 8, 2014 Herbrand's Lemma as a Hasse Diagram  (Seminar presentation)
59 Sep 25, 2013 Contemplations of an octogenarian mathematician  (Lecture Note)
58 July 30, 2013 Drei Frauenschicksale in den Mathematischen Wissenschaften  (Vortragsmanuskript)
57 Jan 24, 2013 Contributions to the History of Number Theory in the 20th Century  (Book, 278 pp.)
56 Dec 22, 2012 What is reciprocity ?  (temporarily not available, will be revised and extended)
55 Oct 23, 2012 Remark on some biographical data of Furtwängler.
54 Oct 12, 2012 Die mathematischen Tagebücher von Helmut Hasse 1923-1925.
53 Aug 12, 2012 Helmut Hasse und die Familie Mendelssohn. (4 pp.)
52 June 18, 2012 The Riemann hypothesis in characteristic p, its origin and development. Part 4. (71 pp.)
51 May 31, 2012 Vortrag über Emmy Noether. (8,66 MB)
50 Jan 11, 2012 A letter from Hasse to Artin. (9 pp.)
49 Jan 10, 2012 Heinrich-Wolfgang Leopoldt 1927-2011. German  version (3pp.) and English (6pp.) .
48. June 17, 2011 Cahit Arf and his invariant. (45 pp.)
47. Sep 12 2010 In Memoriam Ernst Steinitz. (14 pp.)
46. Aug 16, 2010 On the Arf invariant in historical perspective, Part 2. (15 pp.)
45. Mar 19, 2010 Betr. Fotos Emmy Noether.   (+ English translation) 
44. Oct 21, 2009 Mathematik und Kultur.  
43. Sept 22, 2009 On the Arf invariant in historical perspective. (41 pp.)
42. June 10, 2008 Emil Artin - Helmut Hasse. Their correspondence 1923 - 1934.  (Version of Nov 10, 2008)
41. June 10, 2008 Numbers and models, standard and nonstandard.  (Version of Apr 28, 2010) (22 pp.)
40. Mar 27, 2008 Über Wolfgang Wichmann. (Neue Version 18. Apr. 2008.) (4 pp.)
39. Jan 22, 2008 Emmy Noether: Die Gutachten. 
38. May 15, 2007 Emmy Noether und Hermann Weyl.  (39 pp.)
37. Mar 31, 2007 Leserbrief an den Herausgeber der "Mitteilungen der DMV". 
36. Sep 18, 2006 The Riemann hypothesis in characteristic p, its origin and development. Part 3. 
35. May 12, 2006 Helmut Hasse - Emmy Noether. Their correspondence 1925 - 1935. 
34. Mar 12, 2006 Zu Emmy Noethers Geburtstag. (Neue, erweiterte Version 29. Dez. 2006.)  (11 pp.)
33. Feb 25, 2006 Helmut Schaefer 1925-2006.  
32. Mar 9, 2005 From FLT to finite groups. The remarkable career of Otto Grün.  (43 pp.)
31. Jan 21, 2005 Laudatio für B. H. Matzat  
30. Nov 04, 2004 Introduction of Langlands at Arf Lecture   (5.pp.)
29. Feb 28, 2004 The Brauer-Hasse-Noether theorem in historical perspective  
28. Aug 27, 2003 On the embedding problem for global fields   (9 pp.)
27. July 03, 2003 The Riemann hypothesis in characteristic p, its origin and development. Part 2  
26. July 02, 2003 The Riemann hypothesis in characteristic p, its origin and development. Part 1  
25. Feb 13, 2003 Erinnerungen an Helmut Hasse in Berlin (5. Mai 1997)
24. Jan 09, 2003 Reminiscences of Arnold Ross
23. Sep 30, 2002 David Hilbert in Königsberg.  (23 pp.)
22. May 26, 2002 Zum Fermat-Problem (24.1.1998)
21. Apr 16, 2002 Helmut Hasse in Halle  (22 pp.)
20. Mar 25, 2002 David Hilbert (1862-1943). Zum 140. Geburtstag
19. Mar 08, 2002 Albrecht Fröhlich (1916-2001). Obituary
18. Mar 12, 2002 Emmy Noether's contributions to the theory of group rings.
17. Nov 03, 2001 Die Albertina. Über die Königsberger Universitäts-Geschichte (11 pp.)
16. Oct 23, 2001 History of valuation theory. Part I   (73 pp.)
15. Mar 17, 2001 The Brauer-Hasse-Noether theorem in historical perspective
14. Dec 20, 2000 On the theorem of Grunwald-Wang in the setting of valuation theory
13. Nov 26, 2000 On the history of Artin's L-functions and conductors (42 pp.)
12. Nov 18, 2000 August Leopold Crelle  (5 pp.)
11. Jun 09, 2000 Laudatio für Sigrid Böge
10. Dec 06, 1999 Class Field Theory in characteristic p. Its origin and development  (73 pp.)
9. Nov 06, 1998 In memoriam Helmut Hasse
8. Sep 26, 1998 Königsberger Mathematiker im 19. Jahrhundert
7. Dec 15, 1997 Geschichte der Zahlentheorie in den dreißiger Jahren
6. Dec 09, 1997 Einleitung zum Vortrag in Erlangen am 9.12.1997
5. Aug 13, 1997 Exponential sums. The estimate of Hasse-Davenport-Weil  
4. Jan 16, 1997 Bartel Leendert van der Waerden (1903-1996)
3a. July 17, 1996 On convergent power series
3. Sep 01, 1995 Friedrich Wilhelm Bessel
2. Jun 01, 1992 Heinrich Weber, David Hilbert and Königsberg
1. Apr 24, 1987 Vorwort zu: Erich Hecke, Analysis und Zahlentheorie
0. Sep 4, 1987 Über die algebraisch-zahlentheoretischen Arbeiten von Max Deuring.  
-1. 1984 Some Tendencies in Contemporary Algebra.  
-2 1979 Review of Weil, Collected Papers.   (4 pp.)
-3 1995 On Rumely's Local Global Principle.   (32 pp.)

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 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. (2pp.)  

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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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The 20th century was a time of great upheaval and great progress, mathematics not excluded. In order to get the overall picture of trends, developments and results it can be illuminating to look at their manifestations locally, in the personal life and work of people living at the time. The university archives of Göttingen harbor a wealth of papers, letters and manuscripts from several generations of mathematicians -- documents which tell us the story of the historic developments from a local point of view. The present book offers a number of essays based on documents from Göttingen and elsewhere -- those essays which are not yet contained in the author's Collected Works. These little pieces, independent from each other, are meant as contributions to the imposing mosaic of history of number theory. They are written for mathematicians but with no special background requirements. Involved are the names of Emil Artin, Richard Brauer, Helmut Hasse, Emmy Noether, Otto Grün, Hermann Weyl, Abraham Robinson, Cahit Arf, Robert Langlands, Ernst Steinitz, Heinrich-Wolfgang Leopoldt, A.A.Albert, and more.


December 22, 2012 What is reciprocity?  (Temporarily not available, will be revised and extended)

This is a preliminary manuscript which grew out of a talk given on April 15, 2010 at Ovidius University, Constantza at the International Conference "Fundamental Structures of Algebra''. This manuscript is not yet completed. This is the version of July 9, 2013.

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October 23, 2012 Remark on some biographical data of Furtwängler.

The content of these notes has been included into our manuscript "What is reciprocity ?" . See No. 56.

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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.


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 our two previous papers No.43 and 46. For the convenience of the reader we have also included the revised 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 12, 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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August 16, 2010  On the Arf invariant in historical perspective, Part 2.

This article is superseded by the article  No. 48 (Cahit Arf and his invariant).

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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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Oktober 21, 2009 Mathematik und Kultur.

Im März 2009 hatte ich vor einer Diskussionsrunde von interessierten, aber der Mathematik fernstehenden Zuhörern ein Referat zu halten mit dem Titel: "Mathematik und Kultur". Mein Ziel war es, aufzuzeigen, dass die Mathematik in allen uns bekannten Kulturen eine wichtige Rolle gespielt hat und spielt.

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Sep 22, 2009 On the Arf invariant in historical perspective.

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This version is superseded by the article  No.48 (Cahit Arf and his invariant).


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.  

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This is the somewhat extended manuscript of a talk presented at the Hermann Weyl conference in Bielefeld, September 10, 2006. (Corrected and amended version January 28, 2008.) Published in: Groups and Analysis: The legacy of Hermann Weyl, ed. by Karin Tent. London Mathematical Society Lecture Note Series vol. 394 (2008) pp. 285-326

Mar 31, 2007 Leserbrief an den Herausgeber der "Mitteilungen der DMV".  

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Der Text des bekannten Nachrufes von Albert Einstein auf Emmy Noether, der am 5. Mai 1935 in der New York Times erschienen ist, stammt von Hermann Weyl. Dies wird anhand kürzlich aufgefundener Dokumente begründet. Insbesondere wird der Originaltext dieses Nachrufs vorgeführt, der in deutscher Sprache verfasst worden war, bevor er ins Englische übersetzt wurde. (Zusatz am 9.12.2007: Die These, dass der Text des Nachrufes von Weyl stammt, lässt sich nun doch nicht aufrechterhalten. Siehe dazu das Kapitel 8 in dem Artikel Nr.38.)

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

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Abstract: This manuscript is a continuation of Parts 1 and 2. (For those see   July 03, 2003 .) We report on the gradual evolvement and the structure of Hasse's second proof of the Riemann hypothesis for elliptic curves over finite fields. The main feature of this proof is the rebuilding of the theory of complex multiplication by purely algebraic means, thereby making it available for curves over fields of arbitrary characteristic p. We also report on Deuring's subsequent work on the structure of endomorphism rings of elliptic curves in characteristic p, including the supersingular case. Our sources are not only the published papers but also other documents like letters, manuscripts and lecture notes which we have found in several archives. The article is written such that it can be read independently of Parts 1 and 2 of the same series. (65 pages.)

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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Nov 04, 2004   Introduction of Langlands at Arf Lecture

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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.


Feb 28, 2004   The Brauer-Hasse-Noether theorem in historical perspective. (Version of Feb 28, 2004.) "Schriften der Math.-Phys. Klasse der Heidelberger Akademie der Wissenschaften", Nr.15 (2004), 92 p.

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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.)


Aug 27, 2003   On the embedding problem for global fields. Comments to an old paper of Klaus Hoechsmann. Israel Journal of Mathematics (2004) 369-379

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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.


July 3, 2003  The Riemann hypothesis in characteristic p, its origin and development. Part 2. The first steps by Davenport and Hasse. (Version of July 17, 2003). Mitteilungen der Mathematischen Gesellschaft in Hamburg (2004) 5-74

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Abstract: This manuscript is a continuation of Part 1. (For that see   the next entry .) We report on how Hasse met Davenport in 1931, how the latter introduced his friend Hasse to his problem on diophantine congruences, and how Artin identified this problem with the Riemann hypothesis. We discuss Hasse's first proof for elliptic fields which used classic uniformization and complex multiplication; the idea for this developed during a discussion with Mordell about Siegel's great paper on binary diophantine equations.


July 2, 2003   The Riemann hypothesis in characteristic p, its origin and development. Part 1. The formation of the zeta functions of Artin and F.K. Schmidt. (Revised Version of Nov 17, 2004)
Mitteilungen der Mathematischen Gesellschaft in Hamburg, Band 21/2 (2002) 79-157  (This revised version contains several changes against the published version.)


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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.


Feb 13, 2003   Erinnerungen an Helmut Hasse in Berlin.
Manuscript for the foreword to a lecture on Hasse's work, read at the Humboldt University in Berlin on May 5, 1997. 2 pages. (German)
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Abstract: Dies ist das Manuskript für ein Vorwort zu meinem Vortrag am 5. Mai 1997 an der Humboldt-Universität in Berlin. Der Vortrag behandelte die Hasseschen Arbeiten in den dreißiger Jahren zur Riemannschen Vermutung in Funktionenkörpern. Das Vorwort enthält einige persönliche Erinnerungen an Hasse, als ich bei ihm in Berlin in den Jahren 1948-1950 studierte.

Jan 09, 2003   Reminiscences of Arnold Ross.
1 page.
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Abstract: This is the manuscript for a contribution to the Arnold Ross memorial article, edited by Dan Shapiro and Allyn Jackson in the Notices of the American Mathematical Society.

Sep 30, 2002   David Hilbert in Königsberg. (Revised Jan 17, 2003)
23 pages. (German)
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Abstract: This is the manuscript for two lectures delivered to the "Hilbert-Seminar" at Kaliningrad (Königsberg) in Sep/Oct 2002, on the occasion of the 140th birth-year of David Hilbert. It covers the first 33 years of Hilbert's life and work. In those years he grew up in Königsberg and studied at the "Albertina", the University there. He obtained his doctor's degree from the Albertina, did his habilitation and got his first professorship there - before he changed to Göttingen.

May 26, 2002   Zum Fermat-Problem.
Manuscript for a popular lecture at the Mathematics Institute of Heidelberg University, on Jan 24, 1998. 18 pages.(German)
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Am 24.1.1998 fand im Mathematischen Institut der Universität Heidelberg ein "Tag der offenen Tür" statt und ich war kurzfristig aufgefordert worden, dabei einen Vortrag über das Fermat-Problem zu halten. Dies ist das Mansukript zu diesem Vortrag. Die in Fußnoten gesetzten Anmerkungen stammen von Gerhard Frey, dem ich das Manuskript nachträglich vorgelegt hatte.

Abstract: Bereits vor fünf Jahren wurde die Lösung des Fermat-Problems mit großem Pressewirbel angekündigt. Inzwischen ist eine Vielzahl von Artikeln zu diesem Thema erschienen, darunter auch manche, die ich sowohl für Mathematiker als auch für Nichtmathematiker empfehlen kann. Vgl. die Literaturangaben am Schluss. Auch wenn Sie den einen oder den anderen Artikel darüber bereits gelesen haben, so finden Sie vielleicht im folgenden noch einiges, was bisher nicht so bekannt geworden ist.


April 16, 2002   Helmut Hasse in Halle. Jointly with Günther Frei.
In: Martin Goebel, Karin Richter, Kurt Richter (ed.), Aspekte der Mathematikgeschichte in Halle, Report No. 19 (2002) 83-98. (German)
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Abstract: Helmut Hasse (1898-1979) wird der mathematischen Welt als einer der führenden Zahlentheoretiker seiner Zeit in Erinnerung bleiben. Auch über die Zahlentheorie hinaus haben seine Ideen die Mathematik in starkem Maße beeinflusst. Hasse wirkte in Halle in den Jahren 1925-1930. Diese Jahre, zusammen mit den folgenden Jahren 1930-1934 in Marburg, können als die fruchtbarste Periode seines Schaffens angesehen werden. In der folgenden Kurzbiographie Hasses wird insbesondere seine Tätigkeit in Halle herausgestellt. Wir haben eine Reihe von Zitaten aus Briefen und anderen Quellen aufgenommen. Dabei stützen wir uns u.a. auf Dokumente aus dem Archiv der Universität Halle, sowie auf Briefe aus dem Nachlass Hasses, die sich in der Handschriftenabteilung der Universität Göttingen befinden.

March 25, 2002   David Hilbert (1862-1943). Zum 140. Geburtstag
Newspaper article on the occasion of Hilbert's 140th birthday. 5 pages. (German)
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March 8, 2002   Albrecht Fröhlich (1916-2001)
Obituary. Heidelberg Academy of Sciences. 3 pages. (German) (Revised March 20, 2002)
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February 14, 2002   Emmy Noether's contributions to the theory of group rings.
This is a fragment of a manuscript written for a talk given at the Conference "Around Group Rings" at Jasper, Alberta, Feb. 18 - 21, 2002. It is planned to use it, together with our manuscript of March 17, 2001 on the Brauer-Hasse-Noether theorem, as a basis for a larger exposition on the history of the theory of algebras. 13 pages.
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Nov 3, 2001   Die Albertina. Gedanken zu einem neuen Buch über die Königsberger Universitäts-Geschichte.
Manuscript for a news paper article on the book "Albertina" of Kasimir K. Lawrynowicz. 11 pages. (German)
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Oct 23, 2001   History of valuation theory. Part I. (Version of July 23,2003)
In: F.V. Kuhlmann, S. Kuhlmann, M. Marshall (ed.), Valuation Theory and its applications, vol. 1 (Fields Institute Communications, 2002) 291-355
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Abstract: The theory of valuations was started in 1912 by the Hungarian mathematician Josef Kürschák who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of p-adic fields as defined by Kurt Hensel. In the following decades we can observe a quick development of valuation theory, triggered mainly by the discovery that much of algebraic number theory could be better understood by using valuation theoretic notions and methods. An outstanding figure in this development was Helmut Hasse. Independent of the application to number theory, there were essential contributions to valuation theory given by Alexander Ostrowski, published 1934. About the same time Wolfgang Krull gave a more general, universal definition of valuation which turned out to be applicable also in many other mathematical disciplines such as algebraic geometry or functional analysis, thus opening a new era of valuation theory. In the present article which is planned as the first part of more to come, we report on the development of valuation theory until the ideas of Krull about general valuations of arbitrary rank took roots. That is, we cover the pre-Krull era. As our sources we use not only the published articles but also the information contained in letters and other material from that time, mostly but not exclusively from the legacy of Hasse at the University library at Göttingen.

This version contains some minor changes against the published version.


March 17, 2001   The Brauer-Hasse-Noether theorem in historical perspective.
 
This older Version has now been replaced by a new version, revised and extended:   Feb 28, 2004

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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).

 


Dec 20, 2000   On the theorem of Grunwald-Wang in the setting of valuation theory.  
Seminar note, jointly with Falko Lorenz.
In: F.V. Kuhlmann, S. Kuhlmann, M. Marshall (ed.), Valuation Theory and its applications, vol. 2 (Fields Institute Communications, 2002) 43 pages
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Abstract: Given a field K with finitely many valuations; does there exist an extension of K which at these valuations has a prescribed local behavior? The Grunwald-Wang theorem answers this question in the case of abelian field extensions. Originally developed for algebraic number fields in the context of class field theory, it has turned out that it is valid quite generally, for arbitrary multi-valued fields, provided the valuations are of rank one or, more generally, are mutually independent and dense in their respective henselizations. In this paper we present a simple proof which is based on Kummer theory for cyclic Galois algebras, and on Witt theory in case of characteristic p.

Nov 26, 2000   On the history of Artin's L-functions and conductors. Seven letters from Artin to Hasse in the year 1930. (Version July 23, 2003)
Mitteilungen der Mathematischen Gesellschaft in Hamburg, Band  19* (2000) 5-50.
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Abstract: In the year 1923 Emil Artin introduced his new L-functions belonging to Galois characters. But his theory was still incomplete in two respects. First, the theory depended on the validity of the General Reciprocity Law which Artin was unable at that time to prove in full generality. Secondly, in the explicit definition of L-functions the ramified primes could not be taken into account; hence that definition was of provisional character only whereas the final definition could be given in a rather indirect way only. In later years Artin filled both of these gaps: In 1927 he proved the General Reciprocity Law, and in 1930 he gave a complete definition of his L-functions, including the ramified and the infinite primes; at the same time he introduced his theory of conductors for Galois characters. This development is well documented in the correspondence between Artin and Hasse of those years. In the present paper we discuss seven letters from Artin to Hasse, written in the year 1930, where he expounds his ideas about the final definition of the L-functions and about his conductors. We also discuss some letters from Emmy Noether to Hasse of the same time which are directly inspired by Artin's.

This version contains some minor changes against the published version (Jan 10, 2007).


Nov 18, 2000   August Leopold Crelle.
Manuskript einer Rede, gehalten am 18.November 2000 bei der Aufstellung einer Gedenktafel für A.L.Crelle in Eichwerder, seinem Geburtsort. 5 Seiten.
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June 9, 2000   Laudatio für Sigrid Böge.
Manuskript einer Rede, gehalten am 9.Juni 2000 in einem Mathematischen Kolloquium in Heidelberg, aus Anlass der Emeritierung von Professor Sigrid Böge.14 Seiten.
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Dec 6, 1999   Class Field Theory in characteristic p. Its origin and development.
In: K. Miyake (ed.) Class Field Theory - Its Centenary and Prospect (Advanced Studies in Pure Mathematics vol. 30, Tokyo 2000) 549 - 631. (Version July 23,2003)
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Abstract: Today's notion of "global field" comprises number fields (algebraic, of finite degree) and function fields (algebraic, of dimension 1, finite base field). They have many similar arithmetic properties. The systematic study of these similarities seems to have been started by Dedekind (1857). A new impetus was given by the seminal thesis of E.Artin (1921, published in 1924). In this exposition I shall report on the development during the twenties and thirties of our century, with emphasis on the emergence of class field theory for function fields. The names of F.K.Schmidt, H. Hasse, E. Witt, C. Chevalley (among others) are closely connected with that development.

Nov 6, 1998   In Memoriam Helmut Hasse.
Vorwort zu einem Vortrag auf dem Gedenkkolloquium, Hamburg 6.11.1998. 3 Seiten. (German)
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Sep 26, 1998   Königsberger Mathematiker im 19. Jahrhundert.
Text of a talk given at the Franz Neumann seminar at Kaliningrad on Sep 26, 1998. 4 pages. (German). Published in: Rudolf Fritsch et al. ed. Franz Ernst Neumann (1798-1895). Zum 200. Geburtstag des Mathematikers, Physikers und Kristallographen. Verlag "Terra Balitica" Ludwig Maximilians Universität München 2005. (Deutsch-Russische Ausgabe.)
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Dec 15, 1997   Zur Geschichte der Zahlentheorie in den dreißiger Jahren. Die Entstehung der Riemannschen Vermutung für Kurven, und ihres Beweises im elliptischen Fall.
Jahrbuch 1996 der Braunschweigischen Mathematischen Gesellschaft (Göttingen 1997) 153 - 191 
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Abstract: Ich berichte über die Entstehung der Riemannschen Vermutung für Funktionenkörper, oder -- was damit gleichbedeutend ist -- für Kurven, sowie über den ersten Beweis im elliptischen Fall. Dies geschah in den dreißiger Jahren dieses Jahrhunderts, und die Entwicklung wurde entscheidend bestimmt durch die Person von Helmut Hasse, und zwar wesentlich auf Anregung von Harold Davenport und bis zu einem gewissen Grade auch durch L.J.Mordell.

Dec 9, 1997   Einleitung zum Vortrag in Erlangen am 9.12.1997.
Vortrag aus Anlaß der Verleihung des v.Staudt-Preises an Martin Kneser. 6 Seiten. (German)
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Aug 13, 1997   Exponential sums. The estimate of Hasse-Davenport-Weil.
An introductory course, following the 1934 papers by Hasse and Davenport. Notes from the ESCOLA de ALGEBRA, Rio de Janeiro (1996). 29 pages. (Version of July 23,2003)
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This version contains some minor changes against the published version.

Jan 16, 1997   Bartel Leendert van der Waerden (1903--1996).
Obituary, Heidelberg Academy of Sciences. 5 pages. (German)
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July 17, 1996   On convergent power series.
It is proved that the field of convergent Laurent series is existentially closed in the field of formal Laurent series, in arbitrary characteristic p (the base field is assumed to be complete with respect to some non-archimedean rank 1 valuation). The question arose in a correspondence with Franz-Viktor Kuhlmann and Moshe Jarden. 13 pages.
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Sep 1, 1995   Friedrich Wilhelm Bessel. Gedanken zu einer neuen Biographie über den Königsberger Astronomen.
Manuscript for a news paper article on the Bessel biography of Kasimir K. Lavrynovich. 5 pages. (German)
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Jun 1, 1992   Heinrich Weber, David Hilbert and Königsberg
Lecture Notes for a talk delivered at the "Hilbert Seminar" of the University of Kaliningrad in the summer of 1992. 11 pages.
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Apr 24, 1987   Vorwort zu: Erich Hecke, Analysis und Zahlentheorie
Preface to a reprint of notes from a lecture by Hecke, delivered in Hamburg in 1920/21. 10 pages. (German)
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Sep 4, 1987 Über die algebraisch-zahlentheoretischen Arbeiten von Max Deuring.
Dies ist das Manuskript eines Vortrages, gehalten am 15. Mai 1986 im Mathematischen Institut in Göttingen. 17 Seiten.  Jahresbericht der Deutschen Mathematiker-Vereinigung  91 (1989) 109-125
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1984 Some Tendencies in Contemporary Algebra.
This is a contribution to the volume "Perspectives in Mathematics", published on the occasion of the 40-th anniversary of the Oberwolfach Institute by Birkhäuser Verlag (1984).
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1979 Review of Weil, Collected Papers.
This review has appeared in the Jahresbericht der Deutschen Mathematiker Vereinigung 86 (1984), 29-32.
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1994 On Rumely's Local Global Principle (jointly with Barry Green and Florian Pop).
This has appeared in the Jahresbericht der Deutschen Mathematiker Vereinigung 97 (1995), 43-74.
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