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Algebraic functions and projective curves. (English) Zbl 1034.14011

Graduate Texts in Mathematics 215. New York, NY: Springer (ISBN 0-387-95432-5/hbk). xvi, 179 p. (2003).
Index: Chapter 1. Background (Valuations; completions; differential forms; residues; exercises). – Chapter 2. Function fields (Divisors and adeles; Weil differentials; elliptic functions; geometric function fields; residue and duality; exercises). – Chapter 3. Finite extensions (Norm and conorm; scalar extensions; the different; singular prime divisors; Galois extensions; hyperelliptic functions; exercises). – Chapter 4. Projective curves (Projective varieties; maps to \(\mathbb {P}^n\); projective embeddings; Weierstrass points; plane curves; exercises). – Chapter 5. Zeta functions (The Euler product; the functional equation; the Riemann hypothesis; exercises). – Appendix: Elementary field theory
This is a very nice algebraic introduction to the theory of algebraic curves (no geometry) with full, clear and simple proofs. It should be very useful for workers in coding theory.

MSC:

14H05 Algebraic functions and function fields in algebraic geometry
11R42 Zeta functions and \(L\)-functions of number fields
11R58 Arithmetic theory of algebraic function fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G50 Applications to coding theory and cryptography of arithmetic geometry
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