Neron Models - Comprehensive Survey of Neron Models and Construction

In this review of Neron Models the reviewer finds a deep, carefully argued treatment aimed at researchers and advanced students of arithmetic algebraic geometry; the single biggest reason to buy is the book's thorough construction of Neron models from Grothendieck's perspective, which makes it a valuable reference for anyone working on integral structures of abelian varieties. The tone throughout is scholarly and methodical, and the text is best appreciated by readers with a solid background in algebraic geometry who need a rigorous account rather than an introductory overview.

Key Features

• Comprehensive construction: Presents a detailed, step-by-step construction of Neron models using Grothendieck's algebraic geometry framework, which clarifies foundational points for specialists.
• Historical context: Explains that Neron models were introduced by A. Neron in the early 1960s and situates the subject within later developments in arithmetic geometry.
• Focus on integral structure: Emphasizes the integral aspects of abelian varieties over number fields, making it useful for researchers studying arithmetic properties.
• Advanced audience orientation: Written for arithmeticians and algebraic geometers, the exposition assumes familiarity with schemes and related tools, enabling concise yet rigorous arguments.
• Second-part connections: Develops the relationship between Neron models and the relative theories introduced earlier, helping readers apply the construction in broader contexts.

Who It's For

This book is primarily for graduate students, postdocs, and established researchers in algebraic geometry and arithmetic geometry who need a definitive reference on Neron models and their construction. It is especially useful when one requires a precise account rooted in Grothendieck's language and for work on abelian varieties over number fields.

Those seeking an elementary introduction or a gentle pedagogical text on abelian varieties should look elsewhere; the treatment here is not a primer and assumes substantial background in schemes, cohomology, and arithmetic geometry.

Pros & Cons

Pros

• Authoritative, rigorous exposition that reconstructs Neron models in Grothendieck's terms.
• Valuable historical and conceptual framing linking original work by A. Neron to later advances.
• Useful second-part discussion relating models to relative theories, aiding applications.

Cons

• Requires prior advanced background, so it is not accessible as an introductory text for newcomers.

Specifications

Title	Neron Models (Ergebnisse der Mathematik und ihrer Grenzgebiete)
Series	A Series of Modern Surveys in Mathematics
Authors / Editors	Siegfried Bosch, Werner Lutkebohmert, Michel Raynaud
Subject	Neron models, abelian varieties, arithmetic algebraic geometry
Approach	Construction via Grothendieck's algebraic geometry
Intended audience	Arithmeticians and algebraic geometers

Our Verdict

For researchers and advanced students who need a rigorous, Grothendieck-style account of Neron models, this volume is a high-value, long-term reference that clarifies construction and relationships to relative theories; its depth and precision justify the purchase for specialists, while those seeking an elementary introduction should consider more introductory texts first.

Frequently Asked Questions

Does this book cover the original construction by A. Neron?
Yes, the review notes that the volume treats the Neron models introduced by A. Neron in the early 1960s and places them in historical context.

Is the book suitable for beginners?
No, the book assumes significant background in algebraic geometry and is aimed at graduate-level readers and researchers.

What perspective does the book use?
The treatment is given from Grothendieck's algebraic geometry perspective and develops connections to relative theories in a later part of the text.