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Umfang: 1 online resource (xiii, 475 pages) : , digital, PDF file(s).

ISBN: 1-108-13210-3 , 1-108-13378-9 , 1-316-67150-X

Inhalt: This thorough work presents the fundamental results of modular function theory as developed during the nineteenth and early-twentieth centuries. It features beautiful formulas and derives them using skillful and ingenious manipulations, especially classical methods often overlooked today. Starting with the work of Gauss, Abel, and Jacobi, the book then discusses the attempt by Dedekind to construct a theory of modular functions independent of elliptic functions. The latter part of the book explains how Hurwitz completed this task and includes one of Hurwitz's landmark papers, translated by the author, and delves into the work of Ramanujan, Mordell, and Hecke. For graduate students and experts in modular forms, this book demonstrates the relevance of these original sources and thereby provides the reader with new insights into contemporary work in this area.

Anmerkung: Title from publisher's bibliographic system (viewed on 21 Apr 2017). , Cover -- Half title -- Title -- Copyright -- Contents -- Preface -- 1 The Basic Modular Forms of the Nineteenth Century -- 1.1 The Modular Group -- 1.2 Modular Forms -- 1.3 Exercises -- 2 Gauss's Contributions to Modular Forms -- 2.1 Early Work on Elliptic Integrals -- 2.2 Landen and Legendre's Quadratic Transformation -- 2.3 Lagrange's Arithmetic-Geometric Mean -- 2.4 Gauss on the Arithmetic-Geometric Mean -- 2.5 Gauss on Elliptic Functions -- 2.6 Gauss: Theta Functions and Modular Forms -- 2.7 Exercises -- 3 Abel and Jacobi on Elliptic Functions -- 3.1 Preliminary Remarks -- 3.2 Jacobi on Transformations of Orders 3 and 5 -- 3.3 The Jacobi Elliptic Functions -- 3.4 Transformations of Order n and Infinite Products -- 3.5 Jacobi's Transformation Formulas -- 3.6 Equivalent Forms of the Transformation Formulas -- 3.7 The First and Second Transformations -- 3.8 Complementary Transformations -- 3.9 Jacobi's First Supplementary Transformation -- 3.10 Jacobi's Infinite Products for Elliptic Functions -- 3.11 Jacobi's Theory of Theta Functions -- 3.12 Jacobi's Triple Product Identity -- 3.13 Modular Equations and Transformation Theory -- 3.14 Exercises -- 4 Eisenstein and Hurwitz -- 4.1 Preliminary Remarks -- 4.2 Eisenstein's Theory of Trigonometric Functions -- 4.3 Eisenstein's Derivation of the Addition Formula -- 4.4 Eisenstein's Theory of Elliptic Functions -- 4.5 Differential Equations for Elliptic Functions -- 4.6 The Addition Theorem for the Elliptic Function -- 4.7 Eisenstein's Double Product -- 4.8 Elliptic Functions in Terms of the Φ Function -- 4.9 Connection of Φ with Theta Functions -- 4.10 Hurwitz's Fourier Series for Modular Forms -- 4.11 Hurwitz's Proof That ∆(w) Is a Modular Form -- 4.12 Hurwitz's Proof of Eisenstein's Result -- 4.13 Kronecker's Proof of Eisenstein's Result -- 4.14 Exercises. , 5 Hermite's Transformation of Theta Functions -- 5.1 Preliminary Remarks -- 5.2 Hermite's Proof of the Transformation Formula -- 5.3 Smith on Jacobi's Formula for the Product of Four Theta Functions -- 5.4 Exercises -- 6 Complex Variables and Elliptic Functions -- 6.1 Historical Remarks on the Roots of Unity -- 6.2 Simpson and the Ladies Diary -- 6.3 Development of Complex Variables Theory -- 6.4 Hermite: Complex Analysis in Elliptic Functions -- 6.5 Riemann: Meaning of the Elliptic Integral -- 6.6 Weierstrass's Rigorization -- 6.7 The Phragmén-Lindelöf Theorem -- 7 Hypergeometric Functions -- 7.1 Preliminary Remarks -- 7.2 Stirling -- 7.3 Euler and the Hypergeometric Equation -- 7.4 Pfaff's Transformation -- 7.5 Gauss and Quadratic Transformations -- 7.6 Kummer on the Hypergeometric Equation -- 7.7 Riemann and the Schwarzian Derivative -- 7.8 Riemann and the Triangle Functions -- 7.9 The Ratio of the Periods K'/K as a Conformal Map -- 7.10 Schwarz: Hypergeometric Equation with Algebraic Solutions -- 7.11 Exercises -- 8 Dedekind's Paper on Modular Functions -- 8.1 Preliminary Remarks -- 8.2 Dedekind's Approach -- 8.3 The Fundamental Domain for SL[sub(2)] (ℤ) -- 8.4 Tesselation of the Upper Half-plane -- 8.5 Dedekind's Valency Function -- 8.6 Branch Points -- 8.7 Differential Equations -- 8.8 Dedekind's η Function -- 8.9 The Uniqueness of k[sup(2)] -- 8.10 The Connection of η with Theta Functions -- 8.11 Hurwitz's Infinite Product for η(w) -- 8.12 Algebraic Relations among Modular Forms -- 8.13 The Modular Equation -- 8.14 Singular Moduli and Quadratic Forms -- 8.15 Exercises -- 9 The η Function and Dedekind Sums -- 9.1 Preliminary Remarks -- 9.2 Riemann's Notes -- 9.3 Dedekind Sums in Terms of a Periodic Function -- 9.4 Rademacher -- 9.5 Exercises -- 10 Modular Forms and Invariant Theory -- 10.1 Preliminary Remarks. , 10.2 The Early Theory of Invariants -- 10.3 Cayley's Proof of a Result of Abel -- 10.4 Reduction of an Elliptic Integral to Riemann's Normal Form -- 10.5 The Weierstrass Normal Form -- 10.6 Proof of the Infinite Product for ∆ -- 10.7 The Multiplier in Terms of 12√∆ -- 11 The Modular and Multiplier Equations -- 11.1 Preliminary Remarks -- 11.2 Jacobi's Multiplier Equation -- 11.3 Sohnke's Paper on Modular Equations -- 11.4 Brioschi on Jacobi's Multiplier Equation -- 11.5 Joubert on the Multiplier Equation -- 11.6 Kiepert and Klein on the Multiplier Equation -- 11.7 Hurwitz: Roots of the Multiplier Equation -- 11.8 Exercises -- 12 The Theory of Modular Forms as Reworked by Hurwitz -- 12.1 Preliminary Remarks -- 12.2 The Fundamental Domain -- 12.3 An Infinite Product as a Modular Form -- 12.4 The J-Function -- 12.5 An Application to the Theory of Elliptic Functions -- 13 Ramanujan's Euler Products and Modular Forms -- 13.1 Preliminary Remarks -- 13.2 Ramanujan's τ Function -- 13.3 Ramanujan: Product Formula for ∆ -- 13.4 Proof of Identity (13.2) -- 13.5 The Arithmetic Function τ(n) -- 13.6 Mordell on Euler Products -- 13.7 Exercises -- 14 Dirichlet Series and Modular Forms -- 14.1 Preliminary Remarks -- 14.2 Functional Equations for Dirichlet Series -- 14.3 Theta Series in Two Variables -- 14.4 Exercises -- 15 Sums of Squares -- 15.1 Preliminary Remarks -- 15.2 Jacobi's Elliptic Functions Approach -- 15.3 Glaisher -- 15.4 Ramanjuan's Arithmetical Functions -- 15.5 Mordell: Spaces of Modular Forms -- 15.6 Hardy's Singular Series -- 15.7 Hecke's Solution to the Sums of Squares Problem -- 15.8 Exercises -- 16 The Hecke Operators -- 16.1 Preliminary Remarks -- 16.2 The Hecke Operators T(n) -- 16.3 The Operators T(n) in Terms of Matrices λ(n) -- 16.4 Euler Products -- 16.5 Eigenfunctions of the Hecke Operators -- 16.6 The Petersson Inner Product. , 16.7 Exercises -- Appendix: Translation of Hurwitz's Paper of 1904 -- 1. Equivalent Quantities -- 2. The Modular Forms G[sub(n)](ω[sub(1)], ω[sub(2)]) -- 3. The Representation of the Function G[sub(n)] by Power Series -- 4. The Modular Form ∆(ω[sub(1)], ω[sub(2)]) -- 5. The Modular Function J(ω) -- 6. Applications to the Theory of Elliptic Functions -- Bibliography -- Index.

Weitere Ausg.: ISBN 1-107-15938-5

Sprache: Englisch

URL: https://doi.org/10.1017/9781316671504