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**Emil Artin and beyond. Class field theory and \(L\)-functions. With contributions by James Cogdell and Robert Langlands.**
*(English)*
Zbl 1321.11002

Heritage of European Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-146-0/hbk). xiii, 231 p. (2015).

This book contains six chapters, the first four of which were written by the editors. Chapter 1 begins with the quadratic reciprocity law and then presents a summary of the development of class field theory up to Chevalley’s thesis; in addition, it discusses a letter from Chevalley to Hasse as well as Hasse’s answer written in June 1935. The second chapter deals with Artin’s emigration to the US and his situation in Notre Dame, Indiana and Princeton. This is followed by a description of the collaboration between Artin and George Whaples, and finally Chapter 4 gives comments concerning the thesis of Artin’s student Margaret Matchett as well as a transcription of this thesis, which investigates zeta functions of number fields using ideles and is some sort of precursor to Tate’s thesis.

Chapter 5, which was contributed by James Cogdell, discusses Artin’s attempts at finding a path to nonabelian class field theory using his \(L\)-functions; this article is a masterful exposition of the development of Artin’s ideas up to the Langlands conjectures. The final chapter begins with Langlands’ letter to André Weil written in Januay 1967, and is followed by Langlands’ description of how he arrived at his ideas; this section is written in German.

Class field theory and Langlands’ ideas occupy a central part of number theory in the 20th century, and this book is a significant contribution to the history of this development. It is highly recommended to number theorists and may be read with profit by anyone with a good background in algebra and algebraic number theory.

Chapter 5, which was contributed by James Cogdell, discusses Artin’s attempts at finding a path to nonabelian class field theory using his \(L\)-functions; this article is a masterful exposition of the development of Artin’s ideas up to the Langlands conjectures. The final chapter begins with Langlands’ letter to André Weil written in Januay 1967, and is followed by Langlands’ description of how he arrived at his ideas; this section is written in German.

Class field theory and Langlands’ ideas occupy a central part of number theory in the 20th century, and this book is a significant contribution to the history of this development. It is highly recommended to number theorists and may be read with profit by anyone with a good background in algebra and algebraic number theory.

Reviewer: Franz Lemmermeyer (Jagstzell)

### MSC:

11-03 | History of number theory |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11R37 | Class field theory |

11R39 | Langlands-Weil conjectures, nonabelian class field theory |

11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

01A60 | History of mathematics in the 20th century |

01A70 | Biographies, obituaries, personalia, bibliographies |