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Softcover ISBN:  9780821811795 
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Product Code:  CRMM/3.S.B 
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MAA Member Price:  $84.60 $64.35 
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Softcover ISBN:  9780821811795 
Product Code:  CRMM/3.S 
List Price:  $49.00 
MAA Member Price:  $44.10 
AMS Member Price:  $39.20 
eBook ISBN:  9781470438494 
Product Code:  CRMM/3.E 
List Price:  $45.00 
MAA Member Price:  $40.50 
AMS Member Price:  $36.00 
Softcover ISBN:  9780821811795 
eBook ISBN:  9781470438494 
Product Code:  CRMM/3.S.B 
List Price:  $94.00 $71.50 
MAA Member Price:  $84.60 $64.35 
AMS Member Price:  $75.20 $57.20 

Book DetailsCRM Monograph SeriesVolume: 3; 1993; 112 ppMSC: Primary 14; Secondary 11;
The book represents an introduction to the theory of abelian varieties with a view to arithmetic. The aim is to introduce some of the basics of the theory as well as some recent arithmetic applications to graduate students and researchers in other fields. The first part contains proofs of the AbelJacobi theorem, Riemann's relations and the Lefschetz theorem on projective embeddings over the complex numbers in the spirit of S. Lang's book Introduction to algebraic and abelian functions. Then the Jacobians of Fermat curves as well as some modular curves are discussed. Finally, as an application, Faltings' proof of the Mordell conjecture and its intermediate steps, the Tate conjecture and the Shafarevich conjecture, are sketched.
— H. Lange for MathSciNet
ReadershipGraduate students and researchers wishing an introduction to the subject.

Table of Contents

Chapters

Introduction

Chapter 1. Riemann surfaces

Chapter 2. RiemannRoch theorem

Chapter 3. AbelJacobi theorem and period relations

Chapter 4. Divisors and theta functions

Chapter 5. Dimension of the space of theta functions

Chapter 6. Projective embedding and theta functions

Chapter 7. Elliptic curves as the intersection of two quadrics

Chapter 8. The Fermat curve

Chapter 9. Discrete subrgroups of SL(sub 2)(R)

Chapter 10. Riemann surface structure of $\Gamma \setminus \mathcal H^*$

Chapter 11. The modular curve X(N)

Chapter 12. Generalities on abelian varieties

Chapter 13. The conjecture of Tate

Chapter 14. Finiteness of isogeny classes

Chapter 15. Mordell’s conjecture


Reviews

Grew out of a onesemester course given by the author … The notes of this course … have gained a remarkable popularity among students and teachers, mainly for their efficient arrangement, enlightening style, selfcontainedness and—nevertheless manageable—conciseness. The book … preserves the style of these lectures and, in this way, makes them available to a wider class of readers who wish an independent, brief introduction to the subject of abelian varieties.
Zentralblatt MATH


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The book represents an introduction to the theory of abelian varieties with a view to arithmetic. The aim is to introduce some of the basics of the theory as well as some recent arithmetic applications to graduate students and researchers in other fields. The first part contains proofs of the AbelJacobi theorem, Riemann's relations and the Lefschetz theorem on projective embeddings over the complex numbers in the spirit of S. Lang's book Introduction to algebraic and abelian functions. Then the Jacobians of Fermat curves as well as some modular curves are discussed. Finally, as an application, Faltings' proof of the Mordell conjecture and its intermediate steps, the Tate conjecture and the Shafarevich conjecture, are sketched.
— H. Lange for MathSciNet
Graduate students and researchers wishing an introduction to the subject.

Chapters

Introduction

Chapter 1. Riemann surfaces

Chapter 2. RiemannRoch theorem

Chapter 3. AbelJacobi theorem and period relations

Chapter 4. Divisors and theta functions

Chapter 5. Dimension of the space of theta functions

Chapter 6. Projective embedding and theta functions

Chapter 7. Elliptic curves as the intersection of two quadrics

Chapter 8. The Fermat curve

Chapter 9. Discrete subrgroups of SL(sub 2)(R)

Chapter 10. Riemann surface structure of $\Gamma \setminus \mathcal H^*$

Chapter 11. The modular curve X(N)

Chapter 12. Generalities on abelian varieties

Chapter 13. The conjecture of Tate

Chapter 14. Finiteness of isogeny classes

Chapter 15. Mordell’s conjecture

Grew out of a onesemester course given by the author … The notes of this course … have gained a remarkable popularity among students and teachers, mainly for their efficient arrangement, enlightening style, selfcontainedness and—nevertheless manageable—conciseness. The book … preserves the style of these lectures and, in this way, makes them available to a wider class of readers who wish an independent, brief introduction to the subject of abelian varieties.
Zentralblatt MATH