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AuthorLang, Serge. author
TitleIntroduction to Algebraic and Abelian Functions [electronic resource] / by Serge Lang
ImprintNew York, NY : Springer New York, 1982
Edition Second Edition
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Descript X, 170 p. online resource


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 -- ยง1. Lemmas on Valuations -- ยง2. The Riemann-Roch Theorem -- ยง3. Remarks on Differential Forms -- ยง4. Residues in Power Series Fields -- ยง5. The Sum of the Residues -- ยง6. The Genus Formula of Hurwitz -- ยง7. Examples -- ยง8. Differentials of Second Kind -- ยง9. Function Fields and Curves -- ยง10. Divisor Classes -- II The Fermat Curve -- ยง1. The Genus -- ยง2. Differentials -- ยง3. Rational Images of the Fermat Curve -- ยง4. Decomposition of the Divisor Classes -- III The Riemann Surface -- ยง1. Topology and Analytic Structure -- ยง2. Integration on the Riemann Surface -- IV The Theorem of Abel-Jacobi -- ยง1. Abelian Integrals -- ยง2. Abelโ{128}{153}s Theorem -- ยง3. Jacobiโ{128}{153}s Theorem -- ยง4. Riemannโ{128}{153}s Relations -- ยง5. Duality -- V Periods on the Fermat Curve -- ยง1. The Logarithm Symbol -- ยง2. Periods on the Universal Covering Space -- ยง3. Periods on the Fermat Curve -- ยง4. Periods on the Related Curves -- VI Linear Theory of Theta Functions -- ยง1. Associated Linear Forms -- ยง2. Degenerate Theta Functions -- ยง3. Dimension of the Space of Theta Functions -- ยง4. Abelian Functions and Riemann-Roch Theorem on the Torus -- ยง5. Translations of Theta Functions -- ยง6. Projective Embedding -- VII Homomorphisms and Duality -- ยง1. The Complex and Rational Representations -- ยง2. Rational and p-adic Representations -- ยง3. Homomorphisms -- ยง4. Complete Reducibility of Poincare -- ยง5. The Dual Abelian Manifold -- ยง6. Relations with Theta Functions -- ยง7. The Kummer Pairing -- ยง8. Periods and Homology -- VIII Riemann Matrices and Classical Theta Functions -- ยง1. Riemann Matrices -- ยง2. The Siegel Upper Half Space -- ยง3. Fundamental Theta Functions -- IX Involutions and Abelian Manifolds of Quaternion Type -- ยง1. Involutions -- ยง2. Special Generators -- ยง3. Orders -- ยง4. Lattices and Riemann Forms on C2 Determined by Quaternion Algebras -- ยง5. Isomorphism Classes -- X Theta Functions and Divisors -- ยง1. Positive Divisors -- ยง2. Arbitrary Divisors -- ยง3. Existence of a Riemann Form on an Abelian Variety

Mathematics Mathematical analysis Analysis (Mathematics) Mathematics Analysis


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