Hardcover ISBN:  9781470410742 
Product Code:  MMONO/244 
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eBook ISBN:  9781470419608 
Product Code:  MMONO/244.E 
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Hardcover ISBN:  9781470410742 
eBook: ISBN:  9781470419608 
Product Code:  MMONO/244.B 
List Price:  $320.00 $242.50 
MAA Member Price:  $288.00 $218.25 
AMS Member Price:  $256.00 $194.00 
Hardcover ISBN:  9781470410742 
Product Code:  MMONO/244 
List Price:  $165.00 
MAA Member Price:  $148.50 
AMS Member Price:  $132.00 
eBook ISBN:  9781470419608 
Product Code:  MMONO/244.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Hardcover ISBN:  9781470410742 
eBook ISBN:  9781470419608 
Product Code:  MMONO/244.B 
List Price:  $320.00 $242.50 
MAA Member Price:  $288.00 $218.25 
AMS Member Price:  $256.00 $194.00 

Book DetailsTranslations of Mathematical MonographsVolume: 244; 2014; 285 ppMSC: Primary 14; 11; 37;
The main goal of this book is to present the socalled 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 higherdimensional 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.
ReadershipGraduate students interested in Diophantine and Arakelov geometry.

Table of Contents

Chapters

Preliminaries

Geometry of numbers

Arakelov geometry on arithmetic curves

Arakelov geometry on arithmetic surfaces

Arakelov geometry on general arithmetic varieties

Arithmetic volume function and its continuity

NakaiMoishezon criterion on an arithmetic variety

Arithmetic Bogomolov inequality

LangBogomolov conjecture


Additional Material

Reviews

Compared to the earlier books on Arakelov geometry, the current monograph is much more uptodate, detailed, comprehensive, and selfcontained. 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 seventyeight 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 NakaiMoishezon criterion or the arithmetic Bogomolov inequality. This is a timely monograph that should appeal to researchers in this important area of mathematics.
MAA Reviews


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The main goal of this book is to present the socalled 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 higherdimensional 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.
Graduate students interested in Diophantine and Arakelov geometry.

Chapters

Preliminaries

Geometry of numbers

Arakelov geometry on arithmetic curves

Arakelov geometry on arithmetic surfaces

Arakelov geometry on general arithmetic varieties

Arithmetic volume function and its continuity

NakaiMoishezon criterion on an arithmetic variety

Arithmetic Bogomolov inequality

LangBogomolov conjecture

Compared to the earlier books on Arakelov geometry, the current monograph is much more uptodate, detailed, comprehensive, and selfcontained. 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 seventyeight 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 NakaiMoishezon criterion or the arithmetic Bogomolov inequality. This is a timely monograph that should appeal to researchers in this important area of mathematics.
MAA Reviews