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.

Second EditionS. LangIntroduction to Algebraic and Abelian Functions"An excellent and very readable introduction to basic notions in algebraic geometry."-MATHEMATICAL REVIEWS