Introduction to Algebraic and Abelian Functions [Paperback]

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. Abels Theorem.- ?3. Jacobis Theorem.- ?4. Riemanns 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 Tl.