Introduction to Algebraic and Abelian Functions

Introduction to Algebraic and Abelian Functions is a self-contained 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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Main Author: Lang, Serge
Corporate Author: SpringerLink (Online service)
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 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. 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 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