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Elliptic functions

Author: Serge Lang
Publisher: New York : Springer-Verlag, ©1987.
Series: Graduate texts in mathematics, 112.
Edition/Format:   Print book : English : 2nd edView all editions and formats

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Additional Physical Format: Online version:
Lang, Serge, 1927-2005.
Elliptic functions.
New York : Springer-Verlag, ©1987
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Serge Lang
ISBN: 0387965084 9780387965086 3540965084 9783540965084
OCLC Number: 15661205
Description: xi, 326 pages : illustrations ; 25 cm.
Contents: One General Theory.- 1 Elliptic Functions.- 1 The Liouville Theorems.- 2 The Weierstrass Function.- 3 The Addition Theorem.- 4 Isomorphism Classes of Elliptic Curves.- 5 Endomorphisms and Automorphisms.- 2 Homomorphisms.- 1 Points of Finite Order.- 2 Isogenies.- 3 The Involution.- 3 The Modular Function.- 1 The Modular Group.- 2 Automorphic Functions of Degree 2k.- 3 The Modular Function j.- 4 Fourier Expansions.- 1 Expansion for Gk, g2, g3, ? and j.- 2 Expansion for the Weierstrass Function.- 3 Bernoulli Numbers.- 5 The Modular Equation.- 1 Integral Matrices with Positive Determinant.- 2 The Modular Equation.- 3 Relations with Isogenies.- 6 Higher Levels.- 1 Congruence Subgroups.- 2 The Field of Modular Functions Over C.- 3 The Field of Modular Functions Over Q.- 4 Subfields of the Modular Function Field.- 7 Automorphisms of the Modular Function Field.- 1 Rational Adeles of GL2.- 2 Operation of the Rational Adeles on the Modular Function Field.- 3 The Shimura Exact Sequence.- Two Complex Multiplication Elliptic Curves With Singular Invariants.- 8 Results from Algebraic Number Theory.- 1 Lattices in Quadratic Fields.- 2 Completions.- 3 The Decomposition Group and Frobenius Automorphism.- 4 Summary of Class Field Theory.- 9 Reduction of Elliptic Curves.- 1 Non-degenerate Reduction, General Case.- 2 Reduction of Homomorsphisms.- 3 Coverings of Level N.- 4 Reduction of Differential Forms.- 10 Complex Multiplication.- 1 Generation of Class Fields, Deuring's Approach.- 2 Idelic Formulation for Arbitrary Lattices.- 3 Generation of Class Fields by Singular Values of Modular Functions.- 4 The Frobenius Endomorphism.- Appendix A Relation of Kronecker.- 11 Shimura's Reciprocity Law.- 1 Relation Between Generic and Special Extensions.- 2 Application to Quotients of Modular Forms.- 12 The Function ?(??)/?(?).- 1 Behavior Under the Artin Automorphism.- 2 Prime Factorization of its Values.- 3 Analytic Proof for the Congruence Relation of j.- 13 The ?-adic and p-adic Representations of Deuring.- 1 The ?-adic Spaces.- 2 Representations in Characteristic p.- 3 Representations and Isogenies.- 4 Reduction of the Ring of Endomorphisms.- 5 The Deuring Lifting Theorem.- 14 Ihara's Theory.- 1 Deuring Representatives.- 2 The Generic Situation.- 3 Special Situations.- Three Elliptic Curves with Non-Integral Invariant.- 15 The Tate Parametrization.- 1 Elliptic Curves with Non-integral Invariants.- 2 Elliptic Curves Over a Complete Local Ring.- 16 The Isogeny Theorems.- 1 The Galois p-adic Representations.- 2 Results of Kummer Theory.- 3 The Local Isogeny Theorems.- 4 Supersingular Reduction.- 5 The Global Isogeny Theorems.- 17 Division Points Over Number Fields.- 1 A Theorem of Shafarevi?.- 2 The Irreducibility Theorem.- 3 The Horizontal Galois Group.- 4 The Vertical Galois Group.- 5 End of the Proof.- Four Theta Functions and Kronecker Limit Formula.- 18 Product Expansions.- 1 The Sigma and Zeta Function.- Appendix The Skew Symmetric Pairing.- 2 A Normalization and the q-product for the ?-function.- 3 q-expansions Again.- 4 The q-product for ?.- 5 The Eta Function of Dedekind.- 6 Modular Functions of Level 2.- 19 The Siegel Functions and Klein Forms.- 1 The Klein Forms.- 2 The Siegel Functions.- 3 Special Values of the Siegel Functions.- 20 The Kronecker Limit Formulas.- 1 The Poisson Summation Formula.- 2 Examples.- 3 The Function Ks(x).- 4 The Kronecker First Limit Formula.- 5 The Kronecker Second Limit Formula.- 21 The First Limit Formula and L-series.- 1 Relation with L-series.- 2 The Frobenius Determinant.- 3 Application to the L-series.- 22 The Second Limit Formula and L-series.- 1 Gauss Sums.- 2 An Expression for the L-series.- Appendix 1 Algebraic Formulas in Arbitrary Characteristic.- By J. Tate.- 1 Generalized Weierstrass Form.- 2 Canonical Forms.- Appendix 2 The Trace of Frobenius and the Differential of First Kind.- 1 The Trace of Frobenius.- 2 Duality.- 3 The Tate Trace.- 4 The Cartier Operator.- 5 The Hasse Invariant.
Series Title: Graduate texts in mathematics, 112.
Responsibility: Serge Lang.
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Second Edition S. Lang Elliptic Functions "This book is an excellent addition to the literature and provides a rather complete picture of the modern theory of elliptic functions while remaining at a Read more...

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