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 Abel-Jacobi 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

Titles in this series are co-published with the Centre de Recherches Mathématiques.

Readership

Graduate students and researchers wishing an introduction to the subject.

Reviews

"Grew out of a one-semester 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, self-containedness 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

Table of Contents

• Introduction
• Riemann surfaces
• Riemann-Roch theorem
• Abel-Jacobi theorem and period relations
• Divisors and theta functions
• Dimension of the space of theta functions
• Projective embedding and theta functions (chapter 6)
• Elliptic curves as the intersection of two quadrics
• The Fermat curve
• Discrete subrgroups of SL(sub 2)(R)
• Riemann surface structure of $\Gamma \setminus \mathcal H^*$
• The modular curve X(N)
• Generalities on abelian varieties
• The conjecture of Tate
• Finiteness of isogeny classes (chapter 14)
• Mordell's conjecture
• Bibliography
• Index