*(English)*Zbl 1130.11034

The first edition of this outstanding, very comprehensive and nearly sweeping monograph on height functions in modern Diophantine geometry appeared just a little more than one year ago. As for its precise rich contents, its masterly design, and its emphatic appraisal, we may therefore refer to our very recent review of this text (Zbl 1115.11034). The appreciation for this treasure of mathematical writing is certainly best documented by the very fact that, in the meantime, the distinguished “Doob Prize” (2008) for the best recent mathematical monograph has been awarded to the two authors in regard to their book “Heights in Diophantine Geometry”.

In the 2008 Doob Price Citation it is particularly stressed that this treatise masterly combines the various aspects of Diophantine geometry, both from the perspective of arithmetic geometry and of transcendental number theory, and that the choice of subjects is extremely broad. Also, it is emphasized that the text is essentially self-contained, yet surprisingly accessible given the great depth of the material. Finally, it is assessed that the book is a masterpiece regarding its original approach, its incomparable comprehensiveness, its elegance of exposition, its appealing style of writing, and its unrivalled profundity and accuracy.

The present second edition is a paperback reprint of the original (Zbl 1115.11034). However, the authors have taken the opportunity to correct the few minor typing errors in the 2006 original edition, thereby even increasing the already high degree of perfection of their prize-winning book.

It just remains to repeat the meanwhile widely common opinion about this outstanding monograph: “Heights in Diophantine Geometry” by E. Bombieri and W. Gubler is a fundamental and pioneering standard text in the field, which will undoubtedly serve as a basic source for the future development of number theory and arithmetic geometry as a whole.

##### MSC:

11G50 | Heights |

11-02 | Research monographs (number theory) |

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

11G30 | Curves of arbitrary genus or genus $\ne 1$ over global fields |

11G10 | Abelian varieties of dimension $>1$ |

14K15 | Arithmetic ground fields (abelian varieties) |