| Chapter Ⅰ The Riemann-Roch Theorem1. Lemmas on Valuations2. The Riemann-Roch Theorem3. Remarks on Differential Forms4. Residues in Power Series Fields5. The Sum of the Residues6. The Genus Formula of Hurwitz7. Examples8. Differentials of Second Kind9. Function Fields and Curves10. Divisor ClassesChapter Ⅱ The Fermat Curve1. The Genus2. Differentials3. Rational Images of the Fermat Curve 4. Decomposition of the Divisor ClassesChapter Ⅲ The Riemann Surface1. Topology and Analytic Structure2. Integration on the Riemann SurfaceChapter Ⅳ The Theorem of Abel-Jacobi1. Abelian Integrals2. Abel's Theorem3. Jacobi's Theorem4. Riemann's Relations5. DualityChapter Ⅴ Periods on the Fermat Curve1. The Logarithm Symbol2. Periods on the Universal Covering Space3. Periods on the Fermat Curve4. Periods on the Related CurvesChapter Ⅵ Linear Theory of Theta Functions1. Associated Linear Forms2. Degenerate Theta Functions3. Dimension of the Space of Theta Functions4. Abelian Functions and Riemann-Roch Theorem on the Toru5. Translations of Theta Functions6. Projective EmbeddingChapter Ⅶ Homomorphisms and Duality1. The Complex and Rational Representations2. Rational and p-adic Representations3. Homomorphisms4. Complete Reducibility of Poincar5. The Dual Abelian Manifold6. Relations with Theta Functions7. The Kummer Pairing 8. Periods and HomologyChapter Ⅷ Riemann Matrices and Classical Theta Functions1. Riemann Matrices2. The Siegel Upper Half Space3. Fundamental Theta FunctionschapterⅨInvolutions and Abelian Manifolds of Quaternion Type1. Involutions2. Special Gnerators3. Orders4. Lattices and Riemann Forms on Determined by Quaternion Algebras5. Isomorphism ClasseschapteⅩTheta Functions and DivisorsI. Positive Divisors2. Arbitrary Divisors3. Existence of a Riemann Form on an Abelian VarietyBibliographyIndex |
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