Introduction to Algebraic and Abelian Functions

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 Classes.- II The Fermat Curve.-
1. The Genus.-
2. Differentials.-
3. Rational Images of the Fermat Curve.-
4. Decomposition of the Divisor Classes.- III The Riemann Surface.-
1. Topology and Analytic Structure.-
2. Integration on the Riemann Surface.- IV The Theorem of Abel-Jacobi.-
1. Abelian Integrals.-
2. Abel's Theorem.-
3. Jacobi's Theorem.-
4. Riemann's Relations.-
5. Duality.- 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 Curves.- 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 Torus.-
5. Translations of Theta Functions.-
6. Projective Embedding.- VII Homomorphisms and Duality.-
1. The Complex and Rational Representations.-
2. Rational and p-adic Representations.-
3. Homomorphisms.-
4. Complete Reducibility of Poincare.-
5. The Dual Abelian Manifold.-
6. Relations with Theta Functions.-
7. The Kummer Pairing.-
8. Periods and Homology.- VIII Riemann Matrices and Classical Theta Functions.-
1. Riemann Matrices.-
2. The Siegel Upper Half Space.-
3. Fundamental Theta Functions.- IX Involutions and Abelian Manifolds of Quaternion Type.-
1. Involutions.-
2. Special Generators.-
3. Orders.-
4. Lattices and Riemann Forms on C2 Determined by Quaternion Algebras.-
5. Isomorphism Classes.- X Theta Functions and Divisors.-
1. Positive Divisors.-
2. Arbitrary Divisors.-
3. Existence of a Riemann Form on an Abelian Variety.