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 volume is geared toward a second-year graduate course, but it leads naturally to the study of more advanced books listed in the bibliography.
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 volume is geared toward a second-year graduate course, but it leads naturally to the study of more advanced books listed in the bibliography.
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 andRiemann 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.
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 andRiemann 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.
Rezensionen
Second Edition
S. Lang
Introduction to Algebraic and Abelian Functions
"An excellent and very readable introduction to basic notions in algebraic geometry."-MATHEMATICAL REVIEWS
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