Introduction to Algebraic and Abelian Functions

I The Riemann-Roch Theorem.- x1. Lemmas on Valuations.- x2. The Riemann-Roch Theorem.- x3. Remarks on Differential Forms.- x4. Residues in Power Series Fields.- x5. The Sum of the Residues.- x6. The Genus Formula of Hurwitz.- x7. Examples.- x8. Differentials of Second Kind.- x9. Function Fields and Curves.- x10. Divisor Classes.- II The Fermat Curve.- x1. The Genus.- x2. Differentials.- x3. Rational Images of the Fermat Curve.- x4. Decomposition of the Divisor Classes.- III The Riemann Surface.- x1. Topology and Analytic Structure.- x2. Integration on the Riemann Surface.- IV The Theorem of Abel-Jacobi.- x1. Abelian Integrals.- x2. Abel's Theorem.- x3. Jacobi's Theorem.- x4. Riemann's Relations.- x5. Duality.- V Periods on the Fermat Curve.- x1. The Logarithm Symbol.- x2. Periods on the Universal Covering Space.- x3. Periods on the Fermat Curve.- x4. Periods on the Related Curves.- VI Linear Theory of Theta Functions.- x1. Associated Linear Forms.- x2. Degenerate Theta Functions.- x3. Dimension of the Space of Theta Functions.- x4. Abelian Functions and Riemann-Roch Theorem on the Torus.- x5. Translations of Theta Functions.- x6. Projective Embedding.- VII Homomorphisms and Duality.- x1. The Complex and Rational Representations.- x2. Rational and p-adic Representations.- x3. Homomorphisms.- x4. Complete Reducibility of Poincare.- x5. The Dual Abelian Manifold.- x6. Relations with Theta Functions.- x7. The Kummer Pairing.- x8. Periods and Homology.- VIII Riemann Matrices and Classical Theta Functions.- x1. Riemann Matrices.- x2. The Siegel Upper Half Space.- x3. Fundamental Theta Functions.- IX Involutions and Abelian Manifolds of Quaternion Type.- x1. Involutions.- x2. Special Generators.- x3. Orders.- x4. Lattices and Riemann Forms on C2 Determined by Quaternion Algebras.- x5. Isomorphism Classes.- X Theta Functions and Divisors.- x1. Positive Divisors.- x2. Arbitrary Divisors.- x3. Existence of a Riemann Form on an Abelian Variety.