Elliptic and modular functions from Gauss to Dedekind to Hecke:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2017
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | xiii, 475 Seiten Diagramme |
| ISBN: | 9781107159389 |
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| adam_text | Contents
Preface page ix
1 The Basic Modular Forms of the Nineteenth Century 1
1.1 The Modular Group 1
1.2 Modular Forms 5
1.3 Exercises 11
2 Gauss’s Contributions to Modular Forms 13
2.1 Early Work on Elliptic Integrals 13
2.2 Landen and Legendre’s Quadratic Transformation 17
2.3 Lagrange’s Arithmetic-Geometric Mean 18
2.4 Gauss on the Arithmetic-Geometric Mean 20
2.5 Gauss on Elliptic Functions 27
2.6 Gauss: Theta Functions and Modular Forms 32
2.7 Exercises 36
3 Abel and Jacobi on Elliptic Functions 42
3.1 Preliminary Remarks 42
3.2 Jacobi on Transformations of Orders 3 and 5 54
3.3 The Jacobi Elliptic Functions 60
3.4 Transformations of Order n and Infinite Products 63
3.5 Jacobi’s Transformation Formulas 66
3.6 Equivalent Forms of the Transformation Formulas 70
3.7 The First and Second Transformations 71
3.8 Complementary Transformations 72
3.9 Jacobi’s First Supplemental Transformation 74
3.10 Jacobi’s Infinite Products for Elliptic Functions 75
3.11 Jacobi’s Theory of Theta Functions 80
3.12 Jacobi’s Triple Product Identity 86
3.13 Modular Equations and Transformation Theory 89
3.14 Exercises 90
vi Contents
4 Eisenstein and Hurwitz 94
4.1 Preliminary Remarks 94
4.2 Eisenstein’s Theory of Trigonometric Functions 101
4.3 Eisenstein’s Derivation of the Addition Formula 105
4.4 Eisenstein’s Theory of Elliptic Functions 106
4.5 Differential Equations for Elliptic Functions 109
4.6 The Addition Theorem for the Elliptic Function 113
4.7 Eisenstein’s Double Product 115
4.8 Elliptic Functions in Terms of the Փ Function 116
4.9 Connection of Փ with Theta Functions 117
4.10 Hurwitz’s Fourier Series for Modular Forms 123
4.11 Hurwitz’s Proof That A(a ) Is a Modular Form 126
4.12 Hurwitz’s Proof of Eisenstein’s Result 128
4.13 Kronecker’s Proof of Eisenstein’s Result 129
4.14 Exercises 130
5 Hermite’s Transformation of Theta Functions 132
5.1 Preliminary Remarks 132
5.2 Hermite’s Proof of the Transformation Formula 138
5.3 Smith on Jacobi’s Formula for the Product of Four Theta Functions 141
5.4 Exercises 147
6 Complex Variables and Elliptic Functions 149
6.1 Historical Remarks on the Roots of Unity 149
6.2 Simpson and the Ladies Diary 161
6.3 Development of Complex Variables Theory 164
6.4 Hermite: Complex Analysis in Elliptic Functions 172
6.5 Riemann: Meaning of the Elliptic Integral 176
6.6 Weierstrass’s Rigorization 182
6.7 The Phragmen-Lindelof Theorem 184
7 Hypergeometric Functions 188
7.1 Preliminary Remarks 188
7.2 Stirling 189
7.3 Euler and the Hypergeometric Equation 191
7.4 Pfaff’s Transformation 192
7.5 Gauss and Quadratic Transformations 193
7.6 Kummer on the Hypergeometric Equation 196
7.7 Riemann and the Schwarzian Derivative 198
7.8 Riemann and the Triangle Functions 201
7.9 The Ratio of the Periods ^ as a Conformal Map 202
7.10 Schwarz: Hypergeometric Equation with Algebraic Solutions 207
7.11 Exercises 210
8 Dedekind’s Paper on Modular Functions 212
8.1 Preliminary Remarks 212
8.2 Dedekind’s Approach 216
vii
219
222
222
223
225
228
234
234
235
236
238
243
249
251
251
258
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269
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287
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291
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295
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304
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317
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342
Contents
8.3 The Fundamental Domain for SLq. (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 t) Function
8.9 The Uniqueness of k2
8.10 The Connection of r] with Theta Functions
8.11 Hurwitz’s Infinite Product for rj(co)
8.12 Algebraic Relations among Modular Forms
8.13 The Modular Equation
8.14 Singular Moduli and Quadratic Forms
8.15 Exercises
The i] 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
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 A
10.7 The Multiplier in Terms of
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
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 /-Function
12.5 An Application to the Theory of Elliptic Functions
Contents
viii
13 Ramanujan’s Euler Products and Modular Forms 344
13.1 Preliminary Remarks 344
13.2 Ramanujan’s r Function 348
13.3 Ramanujan: Product Formula for A 350
13.4 Proof of Identity (13.2) 353
13.5 The Arithmetic Function r(n) 356
13.6 Mordell on Euler Products 362
13.7 Exercises 367
14 Dirichlet Series and Modular Forms 371
14.1 Preliminary Remarks 371
14.2 Functional Equations for Dirichlet Series 373
14.3 Theta Series in Two Variables 380
14.4 Exercises 382
15 Sums of Squares 384
15.1 Preliminary Remarks 384
15.2 Jacobi’s Elliptic Functions Approach 393
15.3 Glaisher 394
15.4 Ramanjuan’s Arithmetical Functions 397
15.5 Mordell: Spaces of Modular Forms 400
15.6 Hardy’s Singular Series 405
15.7 Hecke’s Solution to the Sums of Squares Problem 410
15.8 Exercises 424
16 The Hecke Operators 426
16.1 Preliminary Remarks 426
16.2 The Hecke Operators Tin) 428
16.3 The Operators Tin) in Terms of Matrices k(n) 434
16.4 Euler Products 438
16.5 Eigenfunctions of the Hecke Operators 439
16.6 The Petersson Inner Product 442
16.7 Exercises 444
Appendix: Translation of Hurwitz’s Paper of 1904 445
§ 1. Equivalent Quantities 445
§2. The Modular Forms G„ia , 448
§3. The Representation of the Function Gn by Power Series 452
§4. The Modular Form A(ct i, 102) 454
§5. The Modular Function J(co) 455
§6. Applications to the Theory of Elliptic Functions 460
Bibliography 453
Index 471
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.
Ranjan Roy is the Huffer Professor of Mathematics and Astronomy at
Beloit College, and has published papers in differential equations, fluid
mechanics, complex analysis, and the development of mathematics. He
has received the Allendoerfer Prize, the Wisconsin MAA teaching award,
and the MAA Haimo Award for Distinguished Mathematics Teaching; he
was twice named Teacher of the Year at Beloit College. He is a co-author
of three chapters in the NIST Handbook of Mathematical Functions, of
Special Functions with Andrews and Askey, and the author of Sources in
the Development of Mathematics.
Cover image: Gauss sculpture by NFN Kalyan.
Photo by Pipo Bonamino.
Cover design by James F. Brisson
Cambridge
UNIVERSITY PRESS
www.cambridge.org
ISBN 978-1-107-15938-9
I
9 781 107 1 59389
|
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| spelling | Roy, Ranjan 1948-2020 Verfasser (DE-588)1022226789 aut Elliptic and modular functions from Gauss to Dedekind to Hecke Ranjan Roy, Beloit College Cambridge Cambridge University Press 2017 xiii, 475 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Geschichte gnd rswk-swf Elliptic functions Modular functions Functions Modulfunktion (DE-588)4039855-9 gnd rswk-swf Elliptische Funktion (DE-588)4134665-8 gnd rswk-swf Elliptische Funktion (DE-588)4134665-8 s Modulfunktion (DE-588)4039855-9 s Geschichte z DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029438529&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029438529&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Roy, Ranjan 1948-2020 Elliptic and modular functions from Gauss to Dedekind to Hecke Elliptic functions Modular functions Functions Modulfunktion (DE-588)4039855-9 gnd Elliptische Funktion (DE-588)4134665-8 gnd |
| subject_GND | (DE-588)4039855-9 (DE-588)4134665-8 |
| title | Elliptic and modular functions from Gauss to Dedekind to Hecke |
| title_auth | Elliptic and modular functions from Gauss to Dedekind to Hecke |
| title_exact_search | Elliptic and modular functions from Gauss to Dedekind to Hecke |
| title_full | Elliptic and modular functions from Gauss to Dedekind to Hecke Ranjan Roy, Beloit College |
| title_fullStr | Elliptic and modular functions from Gauss to Dedekind to Hecke Ranjan Roy, Beloit College |
| title_full_unstemmed | Elliptic and modular functions from Gauss to Dedekind to Hecke Ranjan Roy, Beloit College |
| title_short | Elliptic and modular functions from Gauss to Dedekind to Hecke |
| title_sort | elliptic and modular functions from gauss to dedekind to hecke |
| topic | Elliptic functions Modular functions Functions Modulfunktion (DE-588)4039855-9 gnd Elliptische Funktion (DE-588)4134665-8 gnd |
| topic_facet | Elliptic functions Modular functions Functions Modulfunktion Elliptische Funktion |
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