This volume consists of the English translations of the letters exchanged between Emil Artin to Helmut Hasse written from 1921 until 1958. The letters are accompanied by extensive comments explaining the mathematical background and giving the information needed for understanding these letters. Most letters deal with class field theory and shed a light on the birth of one of its most profound results: Artin's reciprocity law.

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 Goettingen. 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.

The legacy of Helmut Hasse, consisting of letters, manuscripts and other - pers, is kept at theHandschriftenabteilung of the University Library at Gottin- ] gen.Hassehadanextensivecorrespondence;helikedtoexchangemathematical ideas, results and methods freely with his colleagues. There are more than 8000 documents preserved. Although not all of them are of equal mathematical - terest, 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 letterssenttoHasse are availablewhereasmanyoftheletterswhichhadbeensentfromhim, oftenha- written, seem to be lost. So we have to interpolate, as far as possible, from the repliestoHasseandfromothercontexts, inorderto?ndoutwhathehadwritten 1 in his outgoing letters. The present article is largely based on the letters and other documents which I have found concerning the Brauer-Hasse-NoetherTheorem in the theory of algebras; this covers the years around 1931. Besides the do- ments from the Hasse and the Brauer legacy in Gottingen, ] I shall also use some letters from Emmy Noether to Richard Brauer which are preserved at the Bryn Mawr College Library (Pennsylvania, USA)."

Providing the first comprehensive account of the widely unknown cooperation and friendship between Emmy Noether and Helmut Hasse, this book contains English translations of all available letters which were exchanged between them in the years 1925-1935. It features a special chapter on class field theory, a subject which was completely renewed in those years, Noether and Hasse being among its main proponents. These historical items give evidence that Emmy Noether's impact on the development of mathematics is not confined to abstract algebra but also extends to important ideas in modern class field theory as part of algebraic number theory. In her letters, details of proofs appear alongside conjectures and speculations, offering a rich source for those who are interested in the rise and development of mathematical notions and ideas. The letters are supplemented by extensive comments, helping the reader to understand their content within the mathematical environment of the 1920s and 1930s.

Die in diesem Band abgedruckte Vorlesung uber Analysis und Zahlentheorie wurde im Sommersemester 1920 von Erich Hecke an der Universitat Hamburg gehalten. Diese Universitat war kurz zuvor neu gegrundet worden und wurde bald zu einem fuhrenden mathematischen Zentrum, eine Entwicklung, an der Hecke massgebend beteiligt war. Wie in der Wahl des Titels schon zum Ausdruck kommt, knupft Hecke in seiner Vorlesung ganz bewusst an eine grosse, von Dirichlet begru ndete Tradition an. In mancher Hinsicht kann sie als Vorlaufer se ines beruhmten Buches ,,Vorlesungen uber die Theorie der algebrai schen Zahlen" angesehen werden, geht aber teilweise uber jenes hi naus. Das Erscheinen dieses Buches zum 100. Geburtstag Heckes wur digt einen Mathematiker, dessen Werk in letzter Zeit wieder ganz besonders aktuell geworden ist.