Abelian Varieties - Classic Text on Advanced Algebraic Geometry

In this review of Abelian Varieties the bottom line is clear: this is a specialist mathematical monograph best suited to graduate students and researchers who need a focused, classical treatment of topics not fully developed elsewhere. Lang's book has been out of print but remains valuable because it covers concrete constructions and reciprocity results that complement later scheme theoretic works; for readers wanting an accessible account of the Picard and Albanese constructions and explicit applications to Kummer theory, this edition delivers precisely that.

Key Features

• Historical perspective: Lang situates the subject in the developments prior to scheme theory, which helps readers understand motivations behind later formalism.
• Picard variety construction: The book gives a hands-on construction of the Picard variety useful for those studying line bundles and moduli questions.
• Albanese variety treatment: The explicit treatment of the Albanese variety clarifies duality and mapping properties often used in research.
• Reciprocity and Kummer applications: The reciprocity law for correspondences and its application to Kummer theory supply concrete tools for arithmetic and geometric investigations.
• Chow theory for traces and images: Coverage of the K/k-trace and image provides results relevant to practitioners working over nonclosed fields.

Who It's For

Abelian Varieties is ideal for graduate students, number theorists, and algebraic geometers who need detailed constructions and formulae not emphasized in purely scheme theoretic texts. It is especially helpful for researchers who want to see explicit arguments about Picard and Albanese varieties and the use of correspondences in Kummer theory.

Readers seeking a modern, scheme-first textbook may prefer Mumford or later expositions for a different perspective; Lang's book is complementary rather than a replacement and assumes some comfort with classical algebraic geometry language.

Pros & Cons

Pros

• Concise, focused coverage of topics like the Picard and Albanese constructions that many newer texts omit.
• Contains concrete reciprocity results and applications to Kummer theory valuable for arithmetic geometry.
• Useful historical and technical perspective that makes the transition to scheme theoretic treatments clearer.

Cons

• The exposition predates modern scheme language, so readers seeking a fully scheme theoretic approach will need to consult supplementary sources.

Specifications

Our Verdict

Abelian Varieties is a worthwhile purchase for those who need the specific constructions and reciprocity results Lang presents; it complements scheme theoretic texts and remains good value for researchers seeking explicit arguments about Picard and Albanese varieties and Kummer applications.

Frequently Asked Questions

Is this book modern scheme theoretic treatment?
Answer. No, it predates the scheme focus; it provides classical constructions that complement later scheme theoretic works.

Who benefits most from reading this book?
Answer. Graduate students and researchers in algebraic geometry and number theory who want explicit constructions and reciprocity results.

Does the book cover Kummer theory?
Answer. Yes, it includes applications of the reciprocity law for correspondences to Kummer theory.