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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 is to appear in Springer Verlag.

If you are interested then you may read my final book manuscript online
or, if you wish, download it, provided it is used for research only and
not for commercial purposes. If you agree to
this condition then you may click here. Any comment will be welcomed. *
Added
in March,
2019: *Since the book has appeared meanwhile, this manuscript is
not available here any more.

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*.)

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**.*

**2. Sonja Kowalevskaja (1850 - 1891), ***32 pp*.** **
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**3. Sophie Germain (1776-1831), *** in preparation. *
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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.

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.

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

**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 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).*

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. **

*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. ***
*

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:

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

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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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.

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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).

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Dies ist das Manuskript eines Vortrages, gehalten am 15. Mai 1986 im Mathematischen Institut in Göttingen. 17 Seiten.

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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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This review has appeared in the

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This has appeared in the

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This has appeared in the

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This has appeared, jointly with Abraham Robinson, in the

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