代数函数与Abelian函数
2009-8
世界图书出版公司
莱恩
168
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This short book gives an introduction to algebraic and abelian functions, withemphasis on the complex analytic point of view. It could be used for a course or seminar addressed to second year graduate students. The goal is the same as that of the first edition, although I have made a number of additions. I have used the Weil proof of the Riemann-Roch the orem since it is efficient and acquaints the reader with adeles, which are a very useful tool pervading number theory. The proof of the Abel-Jacobi theorem is that given by Artin in a seminar in 1948. As far as I know, the very simple proof for the Jacobi inversion theorem is due to him. The Riemann-Roch theorem and the Abel-Jacobi theorem could form a one semester course. The Riemann relations which come at the end of the treatment of Jacobi's theorem form a bridge with the second part which deals with abelian functionsand theta functions. In May 1949, Weil gave a boost to the basic theory of theta functions in a famous Bourbaki seminar talk. I have followed his exposition of a proof of Poincare that to each divisor on acomplex torus there corresponds a theta function on the universal covering space. However, the correspondence between divisors and theta functions is not needed for the linear theory of theta functions and the projective embedding of the torus when there exists a positive non-degenerate Riemann form. Therefore I have given the proof of existence of a theta function corresponding to a divisor only in the last chapter, so that it does not interfere, with the self-contained treat- ment of the linear theory.
Chapter Ⅰ The Riemann-Roch Theorem 1. Lemmas on Valuations 2. The Riemann-Roch Theorem 3. Remarks on Differential Forms 4. Residues in Power Series Fields 5. The Sum of the Residues 6. The Genus Formula of Hurwitz 7. Examples 8. Differentials of Second Kind 9. Function Fields and Curves 10. Divisor ClassesChapter Ⅱ The Fermat Curve 1. The Genus 2. Differentials 3. Rational Images of the Fermat Curve 4. Decomposition of the Divisor ClassesChapter Ⅲ The Riemann Surface 1. Topology and Analytic Structure 2. Integration on the Riemann SurfaceChapter Ⅳ The Theorem of Abel-Jacobi 1. Abelian Integrals 2. Abel's Theorem 3. Jacobi's Theorem 4. Riemann's Relations 5. DualityChapter Ⅴ Periods on the Fermat Curve 1. The Logarithm Symbol 2. Periods on the Universal Covering Space 3. Periods on the Fermat Curve 4. Periods on the Related CurvesChapter Ⅵ Linear Theory of Theta Functions 1. Associated Linear Forms 2. Degenerate Theta Functions 3. Dimension of the Space of Theta Functions 4. Abelian Functions and Riemann-Roch Theorem on the Toru 5. Translations of Theta Functions 6. Projective EmbeddingChapter Ⅶ Homomorphisms and Duality 1. The Complex and Rational Representations 2. Rational and p-adic Representations 3. Homomorphisms 4. Complete Reducibility of Poincar 5. The Dual Abelian Manifold 6. Relations with Theta Functions 7. The Kummer Pairing 8. Periods and HomologyChapter Ⅷ Riemann Matrices and Classical Theta Functions 1. Riemann Matrices 2. The Siegel Upper Half Space 3. Fundamental Theta Functions……
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