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Algebraic curves over a finite field. (English) Zbl 1200.11042
Princeton Series in Applied Mathematics. Princeton, NJ: Princeton University Press (ISBN 978-0-691-09679-7/hbk). xx, 696 p. (2008).
This book is the first comprehensive account of the theory of algebraic curves in positive characteristic. It synthesizes a vast literature into a well-organized and readable theory. Naturally, the authors have emphasized those aspects which differ from the classical (characteristic zero) theory, as well as those aspects about which they are leading experts. Fortunately this includes many of the most important topics about curves in positive characteristic.
One of the main topics of interest is curves with many rational points over a finite field. The authors emphasize the geometric approach pioneered by Stöhr and Voloch, which studies curves with many points in terms of their embeddings in projective space. Each such embedding yields an upper bound on the number of rational points on the curve; the canonical embedding yields the classical Weil bound, and often there are other embeddings yielding better bounds. Conversely, this approach yields geometric information about curves with many points. The authors treat this theory in great detail in chapters 8, 9, and 10, providing a most welcome introduction to and survey of this powerful tool. The first seven chapters of the book are largely devoted to developing the background results needed for the Stöhr–Voloch theory, and in particular they discuss the properties of curve embeddings in projective space that are unique to positive characteristic.
Another high point of the book is the treatment of curves having large automorphism groups. Chapters 11 and 12 include (among many other things) the first valid proof of the classification announced by Henn of curves of genus \(g\geq 2\) having more than \(8g^3\) automorphisms. These chapters would be valuable merely as the first survey of the large literature on curves with many automorphisms, but they provide much more than that, including many new results and new treatments. This is now required reading for everyone intending to study this topic.
The authors are to be commended for undertaking the tremendous effort which must have gone into this book. They have brought together a staggering amount of material, enabling readers to gain familiarity with both classical and modern tools and results. This book belongs on the shelf of everyone interested in curves in positive characteristic.

11G20 Curves over finite and local fields
11-02 Research exposition (monographs, survey articles) pertaining to number theory
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14G15 Finite ground fields in algebraic geometry
14G17 Positive characteristic ground fields in algebraic geometry
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
14C20 Divisors, linear systems, invertible sheaves
14H25 Arithmetic ground fields for curves