**Graduate Studies in Mathematics**

Volume: 37;
2001;
531 pp;
Hardcover

MSC: Primary 30; 11; 14; 20;

**Print ISBN: 978-0-8218-1392-8
Product Code: GSM/37**

List Price: $90.00

AMS Member Price: $72.00

MAA Member Price: $81.00

**Electronic ISBN: 978-1-4704-2089-5
Product Code: GSM/37.E**

List Price: $84.00

AMS Member Price: $67.20

MAA Member Price: $75.60

# Theta Constants, Riemann Surfaces and the Modular Group: An Introduction with Applications to Uniformization Theorems, Partition Identities and Combinatorial Number Theory

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*Hershel M. Farkas; Irwin Kra*

There are incredibly rich connections between classical analysis and
number theory. For instance, analytic number theory contains many examples of
asymptotic expressions derived from estimates for analytic functions, such as
in the proof of the Prime Number Theorem. In combinatorial number theory, exact
formulas for number-theoretic quantities are derived from relations between
analytic functions. Elliptic functions, especially theta functions, are an
important class of such functions in this context, which had been made clear
already in Jacobi's Fundamenta nova. Theta functions are also
classically connected with Riemann surfaces and with the modular group
\(\Gamma = \mathrm{PSL}(2,\mathbb{Z})\), which provide another path
for insights into number theory.

Farkas and Kra, well-known masters of the theory of Riemann surfaces
and the analysis of theta functions, uncover here interesting
combinatorial identities by means of the function theory on Riemann surfaces
related to the principal congruence subgroups \(\Gamma(k)\). For
instance, the authors use this approach to derive congruences discovered by
Ramanujan for the partition function, with the main ingredient being the
construction of the same function in more than one way. The authors also obtain
a variant on Jacobi's famous result on the number of ways that an integer can
be represented as a sum of four squares, replacing the squares by triangular
numbers and, in the process, obtaining a cleaner result.

The recent trend of applying the ideas and methods of algebraic geometry to
the study of theta functions and number theory has resulted in great advances
in the area. However, the authors choose to stay with the classical point of
view. As a result, their statements and proofs are very concrete. In this
book the mathematician familiar with the algebraic geometry approach to theta
functions and number theory will find many interesting ideas as well as
detailed explanations and derivations of new and old results.

Highlights of the book include systematic studies of theta constant
identities, uniformizations of surfaces represented by subgroups of the modular
group, partition identities, and Fourier coefficients of automorphic
functions.

Prerequisites are a solid understanding of complex analysis, some
familiarity with Riemann surfaces, Fuchsian groups, and elliptic functions, and
an interest in number theory. The book contains summaries of some of the
required material, particularly for theta functions and theta constants.

Readers will find here a careful exposition of a classical point of view of
analysis and number theory. Presented are numerous examples plus suggestions
for research-level problems. The text is suitable for a graduate course or for
independent reading.

#### Readership

Graduate students, research mathematicians interested in complex analysis and number theory.

#### Reviews & Endorsements

Can be useful to experts and novices alike, … details are abundant and developments mainly self-contained, … the book can be read with profit by anyone with a sufficient background in complex analysis, … Farkas and Kra have exposed a great deal of beautiful mathematics, all of it solidly grounded in the classics of our tradition, and yet much of it new. … this elevates their work to a model of exposition … that could be emulated to the benefit of the entire mathematical community.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Theta Constants, Riemann Surfaces and the Modular Group: An Introduction with Applications to Uniformization Theorems, Partition Identities and Combinatorial Number Theory

Table of Contents pages: 1 2

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Introduction xv16 free
- Chapter 1. The modular group and elliptic function theory 126 free
- §1. Möbius transformations 227
- §2. Riemann surfaces 429
- §3. Kleinian groups 530
- 3.1. Generalities 530
- 3.2. The situation of interest 833
- §4. The elliptic paradise 934
- 4.1. The family of tori 934
- 4.2. The algebraic curve associated to a torus 1439
- 4.3. Invariants for tori 2348
- 4.4. Tori with symmetries 2853
- 4.5. Congruent numbers 3156
- 4.6. The plumbing construction 3156
- 4.7. Teiehmüller and moduli spaces for tori 3358
- 4.8. Fiber spaces - the Teiehmüller curve 3358
- §5. Hyperbolic version of elliptic function theory 3762
- 5.1. Fuchsian representation 3863
- 5.2. Symmetries of once punctured tori 4166
- 5.3. The modular group 4368
- 5.4. Geometric interpretations 4570
- 5.5. The period of a punctured torus 4772
- 5.6. The function of degree two on the once punctured torus 4873
- 5.7. The quasi-Fuchsian representation 4873
- §6. Subgroups of the modular group 4974
- 6.1. Basic properties 4974
- 6.2. Poincaré metric on simply connected domains 5075
- 6.3. Fundamental domains 5277
- 6.4. The principal congruence subgroups Γ(k) 5479
- 6.5. Adjoining translations: The subgroups G(k) 6287
- 6.6. The Hecke subgroups Γ[sub(0)](k) 6388
- 6.7. Structure of Γ(k, k) 6590
- 6.8. A two parameter family of groups 6691
- §7. A geometric test for primality 6893

- Chapter 2. Theta functions with characteristics 7196
- §1. Theta functions and theta constants 7297
- 1.1. Definitions and basic properties 7297
- 1.2. The transformation formula 81106
- 1.3. More transformation formulae 87112
- §2. Characteristics 89114
- 2.1. Classes of characteristics 89114
- 2.2. Integral classes of characteristics 93118
- 2.3. Rational classes of characteristics 93118
- 2.4. Invariant classes for Γ(k) 97122
- 2.5. Punctures on H[sup(2)]/Γ(k) and the classes X[sub(0)](k) 98123
- 2.6. The classes in X[sub(0)](k) 100125
- 2.7. Invariant quadruples 105130
- 2.8. Towers 105130
- §3. Punctures and characteristics 106131
- 3.1. A correspondence 106131
- 3.2. Branching 106131
- §4. More invariant classes 107132
- 4.1. Invariant classes for G(k) 107132
- 4.2. Characterization of G(k) 110135
- 4.3. The surface H[sup(2)]/G(k) 112137
- 4.4. Invariant classes for Γ[sub(o)](k) 113138
- 4.5. More homomorphisms 116141
- §5. Elliptic function theory revisited 117142
- 5.1. Function theory on a torus 117142
- 5.2. Projective embeddings of the family of tori 125150
- §6. Conformal mappings of rectangles and Pieard's theorem 126151
- 6.1. Reality conditions 127152
- 6.2. Hyperbolicity and Picard's theorem 128153
- §7. Spaces of N-th order θ-functions 129154
- §8. The Jacobi triple product identity 138163
- 8.1. The triple product identity 138163
- 8.2. The quintuple product identity 143168

- Chapter 3. Function theory for the modular group Γ and its subgroups 147172
- §1. Automorphic forms and functions 148173
- 1.1. Two Banach spaces 148173
- 1.2. Poincaré series 151176
- 1.3. Relative Poincaré series 152177
- 1.4. Projections to the surface 154179
- 1.5. Factors of automorphy 157182
- 1.6. Multiplicative q-forms 159184
- 1.7. Residues 161186
- 1.8. Weierstrass points for subspaces of A(H[sup(2)], G, e) 162187
- §2. Automorphic forms constructed from theta constants 165190
- 2.1. The order of automorphic forms at cusps - Fourier series expansions at l∞ 165190
- 2.2. Automorphic forms for Γ(k) 172197
- 2.3. Meromorphic automorphic functions for Γ(k) 176201
- 2.4. Evaluation of automorphic functions at cusps 177202
- 2.5. Automorphic forms and functions for G(k) 178203
- 2.6. Automorphic forms and functions for Γ[sub(0)](k) 178203
- 2.7. The structure of ⊕[sup(∞)][sub(q=0)]A[sub(q)](H[sup(2)], Γ) and ⊕[sup(∞)][sub(q=0)]A[sub(q)]+(H[sup(2, Γ) 227
- §3. Some special cases (k' = 1) 183208
- 3.1. k = l 183208
- 3.2. k = 2 185210
- 3.3. k = 3 190215
- 3.4. k = 4 198223
- 3.5. k = 5 201226
- 3.6. k = 6 204229
- §4. Primitive invariant automorphic forms 213238
- 4.1. An index 4 subgroup of Γ(k) for even k 213238
- 4.2. A Hilbert space of modified theta constants 215240
- 4.3. Projective representation of Aut H[sup(2)]/Γ(k) 218243
- 4.4. More Hilbert spaces of modified theta constants 223248
- §5. Orders of automorphic forms at cusps 225250
- 5.1. Calculations via Γ[sub(0)](k) 225250
- 5.2. The general case 227252
- §6. The field of meromorphic functions on H[sup(2)]/Γ(k) 228253
- 6.1. Functions of small degree 228253
- 6.2. G(k)-invariant functions 230255
- 6.3. Generators for the function field K(Γ(k)) 231256
- §7. Projective representations 235260
- §8. Some special cases (k' = k) 239264
- 8.1. k = 3 239264
- 8.2. k = 5 242267
- 8.3. The function field for H[sup(2)]/Γ(7) 245270
- 8.4. The projective embedding of H[sup(2)]/Γ(7) 246271
- 8.5. k = 11 248273
- 8.6. k = 13 249274
- 8.7. k = 9 250275
- §9. The function field of H[sup(2)]/Γ(k) over H[sup(2)]/Γ 253278
- §10. Equations that are satisfied by the embedding 253278
- 10.1. The residue theorem 253278
- 10.2. The algorithm 254279
- 10.3. Three term identities 255280
- 10.4. Examples of equations 256281
- §11. Some special cases (restricted characteristics) 260285
- 11.1. Characteristics with m' = k 260285
- 11.2. Characteristics with m = k 260285
- 11.3. Ratios 263288

- Chapter 4. Theta constant identities 265290
- §1. Dimension considerations 268293
- 1.1. The septuple product identity 268293
- 1.2. Further generalizations 272297
- §2. Uniformization considerations 274299
- §3. Elliptic functions as quotients of N-th order theta functions 274299
- 3.1. The Jacobi quartic and derivative formula revisited 274299
- 3.2. More identities - revisited 276301
- 3.3. More identities - new results 277302
- 3.4. More first order applications 279304
- 3.5. Some modular equations 284309
- §4. Identities which arise from modular forms 291316
- 4.1. Multiplicative meromorphic forms 292317
- 4.2. Cusp forms for Γ 294319
- 4.3. Some special results for the primes 5 and 7 297322
- §5. Ramanujan's t-function 297322
- §6. Identities among infinite products 299324
- §7. Identities via logarithmic differentiation 301326
- §8. Averaging automorphic forms 308333
- §9. The groups G(k) 312337

- Chapter 5. Partition theory: Ramanujan congruences 325350
- §1. Calculations of P[Sub(N)](n) 331356
- §2. Some preliminaries 333358
- 2.1. Γ(p, g)-invariant functions 333358
- 2.2. Calculation of divisor of η(N.) 342367
- 2.3. Coset representatives 344369
- §3. Generalities on constructions of Γ[sub(o)](k)-invariant functions 345370
- 3.1. The basic problems 345370
- 3.2. Some generalities 346371
- §4. Constructions of (group) Γ[sub(o)](k)-invariant functions 347372
- 4.1. The direct construction 347372
- 4.2. Averaging Γ(k[sup(n)], k-invariant functions 349374
- 4.3. Bases for K(Γ[sub(o)](k))[sub(0)] and KΓ[sub(o)](k))[sub(∞)] 361386
- 4.4. Precomposing with A[sub(k)] 363388
- §5. Partition identities 369394
- §6. Production of constant functions 375400
- 6.1. The Frobenius automorphism 375400
- 6.2. Constant functions 378403
- 6.3. Congruences 380405
- 6.4. Functions F[sub(k,n,N)] for negative N 383408
- 6.5. Functions F[sub(k-N)] of small degree 386411
- §7. Averaging operators 388413
- 7.1. Automorphisms of K(Γ[sub(0)](k)) 388413
- 7.2. Other linear maps 391416
- §8. Modular equations 392417
- 8.1. k = 2 393418
- 8.2. k = 3 395420
- 8.3. k = 5 396421
- 8.4. k = 7 398423
- 8.5. k = 13 399424
- §9. The ideal of partition identities 399424
- §10. Examples: Calculations for small k 405430
- 10.1. k = 2 405430
- 10.2. k = 3 408433
- 10.3. k = 5 411436
- 10.4. k = 7 413438
- 10.5. k = 11 414439
- 10.6. k = 13 418443
- 10.7. k = 4 419444
- 10.8. k = 6 423448
- §11. The higher level Ramanujan congruences 424449
- 11.1. The level two and three congruences for small primes 424449
- 11.2. The level n congruences for the prime 2 426451
- 11.3. The level n congruences for the prime 5 428453
- 11.4. The level two congruences for the prime 11 430455
- §12. Taylor series expansions for infinite products 430455

- Chapter 6. Identities related to partition functions 439464

Table of Contents pages: 1 2