*(English)*Zbl 0811.11001

[The first ed. (1986) has been reviewed in Zbl 0601.12001.]

Here are the first two sentences of the “Preface for the 2nd edition”: “The principal change in this new edition is a complete rewriting of Chapter XVII on the explicit formulas. Otherwise, I have made a few editions, and a number of corrections.”

For this edition there are two new journal references: (a) *J. Jorgenson* and *S. Lang*, “A Parseval formula for functions with an asymptotic expansion at the origin”, Lect. Notes Math. 1564 (1993; Zbl 0788.30003); and (b) *N.-P. Skoruppa*, “Quick lower bounds for regulators of number fields”, Enseign. Math. 39, 137-141 (1993; Zbl 0803.11060).

There is a very interesting article of *S. Lang* [“Mordell’s review, Siegel’s letter to Mordell’s diophantine geometry, and 20th century mathematics”, Notices Am. Math. Soc. 42, No. 3, 339-350 (1995)]. It reproduces (and discusses extensively) a letter, highly critical, of Siegel in which this book is mentioned – quite derisively. The article makes a fine case for the approach of this volume.

##### MSC:

11-02 | Research monographs (number theory) |

11-01 | Textbooks (number theory) |

11Rxx | Algebraic number theory: global fields |

11Sxx | Algebraic number theory: local and $p$-adic fields |