Abelian Varieties

Serge Lang

Abelian Varieties, based on the work in algebraic geometry done by the Norwegian mathematician Niels Henrik Abel (1802–29), was first published in 1959. It was reissued later in the author Serge Lang’s career without any revisions being made to it. The treatment is still considered a fundamental advanced text in its subject and is designed for mathematics students at the advanced undergraduate and graduate levels. As a prerequisite, you should have some prior experience with elementary qualitative algebraic geometry and familiarity with the elementary theory of algebraic groups.

Because the book concentrates solely on Abelian varieties rather than the more general topic of algebraic groups, the first chapter covers all of the general results on algebraic groups that are pertinent to this subject. A concise introduction is provided at the beginning of each chapter, and a historical and bibliographical remark is presented at the end. Theorems such as the square theorem, divisor classes on an Abelian variety, functorial formulas, the Picard variety of an arbitrary variety, the I-adic representations, and algebraic systems of Abelian varieties are some of the topics covered in this course. An informative appendix that discusses the construction of correspondence serves as the final section of the text.