CRM Monograph Series 1993; 112 pp; softcover Volume: 3 ISBN10: 0821811797 ISBN13: 9780821811795 List Price: US$30 Member Price: US$24 Order Code: CRMM/3.S
 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 Titles in this series are copublished with the Centre de Recherches Mathématiques. Readership Graduate students and researchers wishing an introduction to the subject. 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 andnevertheless manageableconciseness. 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
 RiemannRoch theorem
 AbelJacobi 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
