**Translations of Mathematical Monographs**

2014;
285 pp;
Hardcover

MSC: Primary 14; 11; 37;

Print ISBN: 978-1-4704-1074-2

Product Code: MMONO/244

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

**Electronic ISBN: 978-1-4704-1960-8
Product Code: MMONO/244.E**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

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#### Supplemental Materials

# Arakelov Geometry

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*Atsushi Moriwaki*

The main goal of this book is to present the
so-called birational Arakelov geometry, which can be viewed as an
arithmetic analog of the classical birational geometry, i.e., the
study of big linear series on algebraic varieties. After explaining
classical results about the geometry of numbers, the author starts
with Arakelov geometry for arithmetic curves, and continues with
Arakelov geometry of arithmetic surfaces and higher-dimensional
varieties. The book includes such fundamental results as arithmetic
Hilbert–Samuel formula, arithmetic Nakai–Moishezon criterion,
arithmetic Bogomolov inequality, the existence of small sections, the
continuity of arithmetic volume function, the Lang–Bogomolov
conjecture and so on. In addition, the author presents, with full
details, the proof of Faltings' Riemann–Roch theorem.

Prerequisites for reading this book are the basic results of
algebraic geometry and the language of schemes.

#### Readership

Graduate students interested in Diophantine and Arakelov geometry.

#### Reviews & Endorsements

Compared to the earlier books on Arakelov geometry, the current monograph is much more up-to-date, detailed, comprehensive, and self-contained. The exposition stands out of its high degree of clarity, completeness, rigor and topicality, which also makes the volume an excellent textbook on the subject for seasoned graduate students and young researchers in arithmetic algebraic geometry. The rich bibliography of seventy-eight references certainly serves as a useful guide to further reading with regard to the more recent research literature in the field.

-- Zentralblatt Math

Many important results are presented for the first time in a book, such as the arithmetic Nakai-Moishezon criterion or the arithmetic Bogomolov inequality. This is a timely monograph that should appeal to researchers in this important area of mathematics.

-- MAA Reviews

#### Table of Contents

# Table of Contents

## Arakelov Geometry

- Cover Cover11
- Title page i2
- Contents iii4
- Preface vii8
- Preliminaries 112
- Geometry of numbers 4152
- Arakelov geometry on arithmetic curves 6374
- Arakelov geometry on arithmetic surfaces 8798
- Arakelov geometry on general arithmetic varieties 141152
- Arithmetic volume function and its continuity 197208
- Nakai-Moishezon criterion on an arithmetic variety 225236
- Arithmetic Bogomolov inequality 237248
- Lang-Bogomolov conjecture 249260
- Bibliography 279290
- Index 283294
- Back Cover Back Cover1298