Introduction to Algebraic and Abelian Functions
Introduction to Algebraic and Abelian Functions is a selfcontained presentation of a fundamental subject in algebraic geometry and number theory. For this revised edition, the material on theta functions has been expanded, and the example of the Fermat curves is carried throughout the text. This vo...
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Format:  eBook 
Language:  English 
Published: 
New York, NY
Springer New York
1982, 1982

Edition:  2nd ed. 1982 
Series:  Graduate Texts in Mathematics

Subjects:  
Online Access:  
Collection:  Springer Book Archives 2004  Collection details see MPG.ReNa 
Table of Contents:
 I The RiemannRoch Theorem
 §1. Lemmas on Valuations
 §2. The RiemannRoch 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 AbelJacobi
 §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. Positive Divisors
 §2. Arbitrary Divisors
 §3. Existence of a Riemann Form on an Abelian Variety
 §1. Associated Linear Forms
 §2. Degenerate Theta Functions
 §3. Dimension of the Space of Theta Functions
 §4. Abelian Functions and RiemannRoch 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 padic 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