Contents

Introduction xiii

Chapter 1. The modular group and elliptic function theory 1 §1. Möbius transformations 2 §2. Riemann surfaces 4 §3. Kleinian groups 5 3.1. Generalities 5 3.2. The situation of interest 8 §4. The elliptic paradise 9 4.1. The family of tori 9 4.2. The algebraic curve associated to a torus 14 4.3. Invariants for tori 23 4.4. Tori with symmetries 28 4.5. Congruent numbers 31 4.6. The plumbing construction 31 4.7. Teichmüller and moduli spaces for tori 33 4.8. Fiber spaces – the Teichmüller curve 33 §5. Hyperbolic version of elliptic function theory 37 5.1. Fuchsian representation 38 5.2. Symmetries of once punctured tori 41 5.3. The modular group 43 5.4. Geometric interpretations 45 5.5. The period of a punctured torus 46 5.6. The function of degree two on the once punctured torus 48 5.7. The quasi-Fuchsian representation 48 §6. Subgroups of the modular group 49 6.1. Basic properties 49 6.2. Poincaré metric on simply connected domains 50 6.3. Fundamental domains 52 6.4. The principal congruence subgroups Γ(k) 54 6.5. Adjoining translations: The subgroups G(k) 62 6.6. The Hecke subgroups Γ_{o}(k) 63 6.7. Structure of Γ(k, k) 65 6.8. A two parameter family of groups 66 §7. A geometric test for primality 68

Chapter 2. Theta functions with characteristics 71 §1. Theta functions and theta constants 72 1.1. Definitions and basic properties 72 1.2. The transformation formula 81 1.3. More transformation formulae 88 §2. Characteristics 89 2.1. Classes of characteristics 89 2.2. Integral classes of characteristics 93 2.3. Rational classes of characteristics 93 2.4. Invariant classes for Γ(k) 97 2.5. Punctures on ℍ^{2}/Γ(k) and the classes X_{o}(k) 98 2.6. The classes in X_{o}(k) 100 2.7. Invariant quadruples 105 2.8. Towers 106 §3. Punctures and characteristics 106 3.1. A correspondence 106 3.2. Branching 107 §4. More invariant classes 107 4.1. Invariant classes for G(k) 107 4.2. Characterization of G(k) 110 4.3. The surface ℍ^{2}/G(k) 112 4.4. Invariant classes for Γ_{o}(k) 113 4.5. More homomorphisms 116 §5. Elliptic function theory revisited 117 5.1. Function theory on a torus 117 5.2. Projective embeddings of the family of tori 125 §6. Conformal mappings of rectangles and Picard’s theorem 126 6.1. Reality conditions 127 6.2. Hyperbolicity and Picard’s theorem 129 §7. Spaces of N-th order θ-functions 130 §8. The Jacobi triple product identity 138 8.1. The triple product identity 139 8.2. The quintuple product identity 143

Chapter 3. Function theory for the modular group Γ and its subgroups147 §1. Automorphic forms and functions 148 1.1. Two Banach spaces 148 1.2. Poincaré series 151 1.3. Relative Poincaré series 152 1.4. Projections to the surface 154 1.5. Factors of automorphy 158 1.6. Multiplicative q-forms 159 1.7. Residues 161 1.8. Weierstrass points for subspaces of 𝔸(ℍ^{2}, G, e) 162 §2. Automorphic forms constructed from theta constants 165 2.1. The order of automorphic forms at cusps – Fourier series

expansions at ı∞ 165

2.2. Automorphic forms for Γ(k) 172 2.3. Meromorphic automorphic functions for Γ(k) 176 2.4. Evaluation of automorphic functions at cusps 177 2.5. Automorphic forms and functions for G(k) 178 2.6. Automorphic forms and functions for Γ_{o}(k) 178 2.7. The structure of ⊕_{q=0}^{∞}𝔸_{q}(ℍ^{2}, Γ) and ⊕_{q=0}^{∞}𝔸_{q}^{+}(ℍ^{2}, Γ) 179

§3. Some special cases (k′ = 1) 183 3.1. k = 1 183 3.2. k = 2 185 3.3. k = 3 190 3.4. k = 4 198 3.5. k = 5 202 3.6. k = 6 204 §4. Primitive invariant automorphic forms 213 4.1. An index 4 subgroup of Γ(k) for even k 213 4.2. A Hilbert space of modified theta constants 215 4.3. Projective representation of Aut ℍ^{2}/Γ(k) 218 4.4. More Hilbert spaces of modified theta constants 223 §5. Orders of automorphic forms at cusps 225 5.1. Calculations via Γ_{o}(k) 225 5.2. The general case 227

§6. The field of meromorphic functions on \overline{ℍ^{2}/Γ(k)} 228 6.1. Functions of small degree 228 6.2. G(k)-invariant functions 230 6.3. Generators for the function field K(Γ(k)) 231 §7. Projective representations 236 §8. Some special cases (k′ = k) 240 8.1. k = 3 240 8.2. k = 5 242 8.3. The function field for \overline{ℍ^{2}/Γ(7)} 245 8.4. The projective embedding of \overline{ℍ^{2}/Γ(7)} 246 8.5. k = 11 248 8.6. k = 13 249 8.7. k = 9 250 §9. The function field of \overline{ℍ^{2}/Γ(k)} over \overline{ℍ^{2}/Γ} 253 §10. Equations that are satisfied by the embedding 254 10.1. The residue theorem 254 10.2. The algorithm 254 10.3. Three term identities 255 10.4. Examples of equations 257 §11. Some special cases (restricted characteristics) 260 11.1. Characteristics with m′ = k 261 11.2. Characteristics with m = k 261 11.3. Ratios 264

Chapter 4. Theta constant identities 265 §1. Dimension considerations 268 1.1. The septuple product identity 268 1.2. Further generalizations 272 §2. Uniformization considerations 274 §3. Elliptic functions as quotients of N-th order theta functions 274 3.1. The Jacobi quartic and derivative formula revisited 274 3.2. More identities - revisited 276 3.3. More identities - new results 277 3.4. More first order applications 279 3.5. Some modular equations 284 §4. Identities which arise from modular forms 291 4.1. Multiplicative meromorphic forms. 292 4.2. Cusp forms for Γ. 294 4.3. Some special results for the primes 5 and 7 297 §5. Ramanujan’s τ-function 297 §6. Identities among infinite products 299 §7. Identities via logarithmic differentiation 301 §8. Averaging automorphic forms 308 §9. The groups G(k) 312 Chapter 5. Partition theory: Ramanujan congruences 325

§1. Calculations of P_{N}(n) 331 §2. Some preliminaries 333 2.1. Γ(p, q)-invariant functions 333 2.2. Calculation of divisor of η(N·) 342 2.3. Coset representatives 344 §3. Generalities on constructions of Γ_{o}(k)-invariant functions 345 3.1. The basic problems 345 3.2. Some generalities 346 §4. Constructions of (group) Γ_{o}(k)-invariant functions 347 4.1. The direct construction 347 4.2. Averaging Γ(k^{n}, k)-invariant functions 349 4.3. Bases for K(Γ_{o}(k))_{0} and K(Γ_{o}(k))_{∞} 360 4.4. Precomposing with A_{k} 363 §5. Partition identities 369 §6. Production of constant functions 375 6.1. The Frobenius automorphism 376 6.2. Constant functions 378 6.3. Congruences 380 6.4. Functions F_{k,n,N} for negative N 384 6.5. Functions F_{k,−N} of small degree 387 §7. Averaging operators 389 7.1. Automorphisms of K(Γ_{o}(k)) 389 7.2. Other linear maps 392 §8. Modular equations 392 8.1. k = 2 393 8.2. k = 3 395 8.3. k = 5 397 8.4. k = 7 399 8.5. k = 13 399 §9. The ideal of partition identities 400 §10. Examples: Calculations for small k 405 10.1. k = 2 405 10.2. k = 3 409 10.3. k = 5 412 10.4. k = 7 414 10.5. k = 11 415 10.6. k = 13 419 10.7. k = 4 420 10.8. k = 6 423 §11. The higher level Ramanujan congruences 425 11.1. The level two and three congruences for small primes 425 11.2. The level n congruences for the prime 2 427 11.3. The level n congruences for the prime 5 428 11.4. The level two congruences for the prime 11 431 §12. Taylor series expansions for infinite products 431

Chapter 6. Identities related to partition functions 439 §1. Some more identities related to covering maps 439 1.1. k = 2 440 1.2. k = 3 440 1.3. k = 5 442 1.4. k = 7 443 1.5. k = 11 444 §2. The ȷ-function and generalizations of the discriminant Δ 447 §3. Congruences for the Laurent coefficients of the ȷ-function 452 3.1. Averaging f_{k}^{i} 457 3.2. Completion of the proof of Theorem 3.6 for k = 5 458 3.3. Proof of Theorem 3.6 for k = 11, n = 1 459 3.4. A further analysis of the k = 2 case 461

Chapter 7. Combinatorial and number theoretic applications 463 §1. Generalities on partitions 464 1.1. Euler series and some old identities 467 1.2. Partitions and sums of divisors 472 1.3. Lambert series 473 §2. Identities among partitions 481 2.1. A curious property of 8 481 2.2. A curious property of 3 481 2.3. A curious property of 7 482 §3. Partitions, divisors, and sums of triangular numbers 482 3.1. Sums of 4 squares 487 3.2. A remarkable formula 489 3.3. Weighted sums of triangular numbers 494 §4. Counting points on conic sections 496 §5. Continued fractions and partitions 500 §6. Return to theta functions 504

Bibliography 511

Bibliographical Notes 513

Index 527

Introduction

The theory of compact Riemann surfaces brings together diverse areas of mathematics. Its building blocks include vast areas of analysis (including Lie theory), geometry/topology and algebra. This was our point of view in our book on Riemann surfaces [6] and it dictated the material to be included in that volume. In particular, we presented a modern approach to the theory of compact Riemann surfaces based on classical methods that prepared the reader to study the modern theories of moduli of surfaces. In this book we head in a different direction and develop another classical connection: to combinatorial number theory. We do not neglect, however, the connections to the problem of uniformizing surfaces represented by very special Fuchsian groups. Problems in number theory can be reformulated as questions about Riemann surfaces, and many of the answers to some of these questions are obtained using function theory. Even though it is an old idea to use function theory (compact Riemann surfaces and automorphic forms) to study analytic and combinatorial number theory and there are many results in these fields, we found it hard to dig out the underlying function theory in the publications of number theorists. No doubt, this is our failing. But since others may also have a deficiency in this area, we decided to organize the material from this point of view. There is new material in this book that has not previously appeared in print, and part of our aim is to present this material to as wide an audience as possible. Our more important aim, however, is to expose the reader to a beautiful chapter in function theory and its applications.

The main actors in our presentation are genus one theta functions and theta constants^{1} (including the classical η-function), the modular group Γ = PSL(2, ℤ), and some of the Riemann surfaces that arise as quotients of the action of finite index subgroups of Γ on ℍ^{2}. We are particularly interested in the principal congruence subgroups Γ(k) and the related subgroups Γ_{o}(k) for (usually small) primes k. Some very interesting combinatorial identities follow from the function theory on these surfaces.^{2}

Theta functions and theta constants with integral characteristics are classical objects intimately connected with the principal congruence subgroup of level 2, Γ(2). This theory is well understood and has as one of its consequences the theorem of Picard: every entire function which omits two values is constant. As is well known, the basic ingredients in the proof of Picard’s theorem are that the holomorphic universal covering of the sphere punctured at three points is the upper half plane and that its fundamental group is Γ(2). We use theta constants with even integral characteristics to construct the universal covering map, and in this way obtain, without using the general uniformization theorem, the hyperbolicity of the three times punctured sphere. The universal covering map is constructed here as a quotient of fourth powers of any two of the three theta constants. We noticed that in this construction the three even characteristics correspond in a natural way to the three punctures on ℍ^{2}/Γ(2), and we began to wonder about natural generalizations. In this book, we present the answer to these inquiries. We uniformize the Riemann surfaces ℍ^{2}/Γ(k) using theta constants with special rational characteristics, and establish a one-to-one (almost canonical) correspondence between the punctures on ℍ^{2}/Γ(k) and certain equivalence classes of characteristics. For example, the four punctures on ℍ^{2}/Γ(3) correspond to the characteristics

[ 1_{\frac{1}{3}} ] , [ \frac{1}{3} \frac{1}{3} ] , [ 1^{\frac{1}{3}} ] , [ \frac{1}{3} \frac{5}{3} ] .

Furthermore, the Riemann surface is uniformized by a quotient of cubes of any two of the four corresponding theta constants. Similar, but obviously more complicated, expressions uniformize the surfaces represented by the higher level congruence groups. Multiple uniformizations of the same Riemann surface lead to theta identities. It is an open problem to determine uniformizations of all four punctured spheres by the methods described above.

1Thus the θ-functions we study, θ[χ](ζ, τ), depend on three variables: a characteristic χ ∈ ℝ^{2}; a variable ζ ∈ ℂ; and a parameter τ ∈ ℍ^{2}, the upper half plane. Fixing the variable ζ = 0 yields the family of theta constants, an abuse of notation since these are holomorphic functions on ℍ^{2}; as functions of the local coordinates q = e^{2πıτ} these are classically known as q-series. We will use the symbol x for the local variable, since tradition in (parts of complex analysis) reserves the letter q for the weight of an automorphic form.

2What is interesting is clearly in the eyes of the beholder. The identities we discuss are obviously interesting to us. The reader must decide whether or not to share our enthusiasm.

The theta constants which appear in our constructions of the uniformizing functions for ℍ^{2}/Γ(3) are closely related with the formulae used by Euler and Ramanujan in the theory of partitions. Specifically, we note that^{3}

θ [ 1^{\frac{1}{3}} ] (0, τ) = exp (\frac{πı}{6}) x^{\frac{1}{24}} ∑_{n=−∞}^{∞}(−1)^{n}x^{\frac{3n^{2}+n}{2}} = exp (\frac{πı}{6}) x^{\frac{1}{24}} ∏_{n=1}^{∞}(1−x^{n}),

where x = exp (\frac{2πıτ}{3} ). Continuing in this direction, we discover that uniformizations of the Riemann surfaces ℍ^{2}/Γ(k) involve functions which appear in the Jacobi triple product. We give a function theoretic proof of this famous formula and then generalize it to the quintuple and septuple product identities, explaining along the way why the formulae obtained are natural from the point of view of the theory of N-th order theta functions. The highlights of the book are systematic studies of theta constant identities, uniformizations of surfaces represented by subgroups of the modular group, partition identities and Fourier series coefficients of automorphic functions, and identities involving the σ-function and Fourier series coefficients of automorphic forms. More detailed information on the contents of each of the chapters follows.

In Chapter 1 we explain the genesis of the modular group in our theory. This group appears naturally when one classifies compact Riemann surfaces of genus one (elliptic curves) up to conformal equivalence. We discuss the generators of this group, find all the fixed points of elements of this group and describe some of the subgroups we shall need in the sequel. Almost everything we do in this chapter is well known and covered in a standard course on complex variables. We describe the structure of the Riemann surfaces ℍ^{2}/G for subgroups G of PSL(2, ℤ). In order to show that this well known and elementary material has nontrivial consequences, we use this theory to show that factors of integers of the form N^{2} + 1 are always the sums of two squares, and we give a geometric criterion for N^{2} + 1 to be a prime number. The result is that N^{2}+1 is prime if and only if the portion in the upper half plane of the straight line joining the origin to the point N + ı in the complex plane intersects the orbit of ı under PSL(2, ℤ) in exactly two points, namely N + ı and \frac{N+ı}{N^{2}+1}. We have included some of the function theoretic prerequisites in this chapter. However, most of the prerequisites will be described when needed. In general, we provide full definitions of all concepts. We do not repeat proofs or arguments readily available in other books, but do reproduce, usually in modified form, proofs from many research papers. The bibliography of relevant books (after the last chapter) is followed by a set of bibliographical notes containing an (incomplete) list of research and expository notes on the material covered by this volume.

3The reader may at this point conclude that the η-function is a disguised theta constant with a rational characteristic. It will also become obvious that the prime 3 plays a special role in our drama.

In Chapter 2 we define the theta functions and theta constants with

characteristics and specialize to rational characteristics of the form [ \frac{m}{k} ; \frac{m′}{k} ]

with m, m′ and k integers^{4} of the same parity in order to construct a correspondence between equivalence classes of sets of characteristics and the punctures on the surface ℍ^{2}/Γ(k). In this chapter we derive a most important property of the theta functions and theta constants, the transformation formula (a significant generalization of known transformation rules) for the action of PSL(2, ℤ) on the upper half plane, and we give a function theoretic proof of the Jacobi triple product formula and some generalizations. The transformation formula allows us to use theta functions to construct modular and cusp forms for subgroups of PSL(2, ℤ). The function theoretic proof of the Jacobi triple product formula yields new proofs of important identities of Jacobi and Euler that are needed for our presentation of partition theory in Chapter 5. We construct theta constant identities which turn out to agree with discoveries of Ramanujan. Our derivations of these identities are on the one hand quite natural, and on the other hand lead to simpler expressions of the equivalent identities discovered by Ramanujan in the sense that they do not involve irrationalities (extracting roots of single valued functions) until they are artificially introduced. It appears that the theta constants which we use are a lot richer than the ones that Ramanujan had at his disposal.

Chapter 3, in a sense, contains the most important material of the book. In it we construct automorphic forms and functions for the principal congruence subgroups and some related groups. The theory we describe is particularly well suited for the study of ℍ^{2}/Γ(k), and we obtain holomorphic mappings of these Riemann surfaces into projective spaces of rather low dimensions. Some interesting geometry and topology emerges as we observe connections of the principal congruence subgroups with the Platonic solids. This phenomenon first occurs for k = 3, 4 and 5. In these cases, Γ/Γ(k) ≅ PSL(2, ℤ_{k}) are the symmetry groups of the regular tetrahedron, octahedron and icosahedron, respectively. This suggests a relation between the images of these curves in the projective space and the regular solids and leads to a generalization of the regular solids based on curves of (some) positive genera. While our development is most suited for the groups Γ(k), for many of the most important applications we need to construct automorphic forms for Γ_{o}(k). Part of the extra difficulties involves the presence of torsion in these groups. We need more detailed analysis, in this and subsequent chapters, to handle these richer groups.

4Assume, unless otherwise stated, for these introductory remarks that k is a (positive) prime.

Chapter 4 is a systematic study of theta identities. Theta constant identities are interesting for several reasons. One reason is their inherent elegance and symmetry. There is something tantalizingly beautiful about the identities of Jacobi, for example,

θ^{2} [ \begin{matrix}0 \\ 0 \end{matrix} ] (0, τ)θ^{2} [ \begin{matrix}0 \\ 0 \end{matrix} ] (z, τ)

= θ^{2} [ \begin{matrix}0 \\ 1 \end{matrix} ] (0, τ)θ^{2} [ \begin{matrix}0 \\ 1 \end{matrix} ] (z, τ) + θ^{2} [ \begin{matrix}1 \\ 0 \end{matrix} ] (0, τ)θ^{2} [ \begin{matrix}1 \\ 0 \end{matrix} ] (z, τ)

or its restriction to z = 0 (known as the Jacobi quartic identity)

θ^{4} [ \begin{matrix}0 \\ 0 \end{matrix} ] (0, τ) = θ^{4} [ \begin{matrix}0 \\ 1 \end{matrix} ] (0, τ) + θ^{4} [ \begin{matrix}1 \\ 0 \end{matrix} ] (0, τ).

Aside from the inherent beauty of the form there is a combinatorial content to the identity. It relates the number of representations of an integer as a sum of four squares to its representation as a sum of four triangular numbers. This is of course just the beginning of a chapter. As one delves deeper into the theory, one finds more and more beautiful identities with more and more combinatorial content.

We present four distinct ways of constructing such identities. In two of theses methods, we use the classical technique of constructing finite dimensional linear spaces of theta functions or modular forms or functions on certain Riemann surfaces and use linear algebra and the simple notions of independence and dependence. We present another newer technique which makes use of the fact that we can use theta functions to construct elliptic functions and the fact that the sum of the residues of an elliptic function in a period parallelogram vanishes. This technique is very powerful and succeeds in giving a rather large set of identities. The main idea here is to construct the correct elliptic function, which turns out to be more an art than a science, that leads to an interesting identity. The fourth method for constructing identities uses uniformizations of Riemann surfaces. In a very nontrivial sense the next two chapters are also studies of theta identities, this time of a very special sort with a very special purpose.

In Chapter 5 we turn to the congruences discovered by Ramanujan for the partition function and show how they follow in a rather simple way from function theory on the appropriate Riemann surfaces. The main ingredient is the construction of the same function in more than one way. Some of the constructions involve averaging operators. It turns out that the averaging processes produce in some cases constant functions. We study the many implications of these constructions, especially of the appearance of constant functions. This material may not be new to the literature. We present it in a unified way based on function theoretic foundations that most of the time remove and in general isolate the mysteries in many of the research monographs on the subject. This chapter is based almost entirely on the properties of the classical η-function, a very special case of the theory described in the previous chapters. We need to know rather detailed information on its multiplier system. The needed number theoretic arguments are found in [16], for example. We would like to avoid dependence on these. We have only partially succeeded in doing so and have hence not included in this volume most of the results of this effort.

In Chapter 6 we begin by reviewing some concepts from covering space theory and show how these ideas lead to beautiful identities among theta constants and their interpretation as identities among infinite products. Here the main tools are the uniformizations of the Riemann surfaces in question. We then continue by showing how many of the ideas used in the congruences related with the partition function and its generalizations carry over to other modular forms. We treat in particular the ȷ-function and the congruences satisfied by the coefficients of its Laurent series expansion.

In Chapter 7 we show how statements about partitions are related to other combinatorial quantities such as representations of positive integers as sums of squares or of triangular numbers, and most importantly to the divisors of an integer. In particular, we describe relations to the question of primality of integers depending on statements about partitions. This suggests that while primality is usually thought of as a subject in multiplicative number theory, it can also be viewed as a part of additive number theory.

We give some examples to show what type of results can be expected in this chapter in the expectation that these applications are the main interests of some readers of this text. We emphasize that these results were not the reason for writing this book. The list of examples is by no means exhaustive. Let σ(n) denote the classical σ-function; that is, σ(n) is the sum of the divisors of the positive integer n. We show that

∑_{j=−∞}^{∞} σ (n − \frac{3j^{2} + j}{2} ) = 0

whenever n is not of the form \frac{3m^{2}+m}{2} with m ∈ ℤ. A companion related result is

∑_{j=0}^{∞} σ (n − \frac{j^{2} + j}{2} ) = 4 ∑_{j=0}^{∞} σ (\frac{n}{2} + \frac{j^{2} + j}{4} )

whenever n is not of the form \frac{m^{2}+m}{2} with m ∈ ℤ. We obtain a variant of Jacobi’s result on the number of ways a positive integer can be written as a sum of four squares by replacing squares with triangular numbers, obtaining, in our view, a cleaner result. A last example of the type of result we will discuss in this chapter is the following. Let S be a set consisting of the positive integers with an additional copy of those positive integers congruent to zero modulo 7. Decompose S into its even and odd parts, E and O respectively. Denote by P_{E}(k) and P_{O}(k) the number of partitions of k with parts taken from the sets E and O respectively. We prove that for all nonnegative integers k,

P_{E}(2k) = P_{O}(2k + 1).

The prerequisites for this book are a thorough understanding of material traditionally covered in first year graduate courses – especially the contents of the complex analysis course. We review, however, the most salient points about elliptic function theory portions of this course. Although a knowledge of Riemann surfaces and Fuchsian groups is helpful, it is not needed by the reader who is willing to accept the summaries of the required material (with references to the literature). Although we do not, in general, reproduce material available in other textbooks, we make an exception for material on theta functions and theta constants despite the availability of excellent sources (for example, [23]). We do so, not only for the convenience of the reader, but also to emphasize our point of view. We have also ignored, to a great extent, the combinatorial and special functions connection. These are discussed in [2], [7] and [10], for example.

A road map While we did not intend to write an encyclopedic text, the result has been quite a large book. We take the liberty of offering the readers our suggestions for possible ways of going through this text, which was written with several different types of readers in mind. These range from the beginning graduate mathematics student through the professional mathematician whose interests are either in combinatorial mathematics (partition theory, representation as sums of squares, counting points on conic sections) or function theory (Riemann surfaces, modular forms). Theoretical physicists might be interested in portions of the material we cover.

The reader is expected to have a reasonable knowledge of the theory of functions of a complex variable, through the Riemann mapping theorem, and enough mathematical maturity to follow an argument even though unfamiliar with the proofs of all the tools used. Thus, the book can be used as a text for a topics course in either analysis or analytic number theory, and as such a reasonable approach would be to go through Chapter 1 sections 1-4.5 and sections 6 and 7. The above material is essentially background material to acquaint the reader with the domains on which we will be doing analysis to obtain the combinatorial results. This reader should then continue with Chapter 2, where the theory of the one dimensional theta function is presented. This chapter should be read in its entirety. In Chapter 3 the reader or instructor pressed for time could read quickly the first section for the definitions and then go on to a careful study of sections 2 through 8.4. Chapter 4 should be read in its entirety. The above could constitute a one semester topics course for beginning graduate students.

The above suggestion leaves out Chapters 5, 6 and 7, which occupy much of this book and are an important part of it, since they deal with the theory of congruences for the Ramanujan partition function, the congruences for the ȷ-function, and the combinatorial interpretations of many of the identities derived in Chapter 4. In a one-year course the material studied could include Chapter 5 through section 10.8 and section 12,^{5} all of Chapters 6 and 7.

The professional mathematician who is interested in Riemann surface theory should study Chapter 1, including sections 4.6 through 5.7, in order to get a picture of where the theory can possibly go. If conversant with the theory of the Riemann theta function, the reader can skip section 1 of Chapter 2 and read the remainder of that chapter. The reader should then proceed to the beginning of Chapter 3. Some of the introductory material of this chapter can be skipped or read quickly to get acquainted with the notation used; the choice of which of the remaining sections to read should be guided by interests; we suggest that this include section 10. Chapter 4 should be read in its entirety and then Chapter 5 and Chapter 6, once again guided by the interests of the reader. Chapter 7 should also be read in its entirety.

The professional mathematician whose interests are in combinatorial mathematics may wish to begin by looking at Chapter 7 and then proceed backwards through the theory. Needless to say, Chapters 2, 3 and 4 will have to be read at some point, and if interested in Ramanujan congruences, Chapter 5 is a must. In any event section 12 of Chapter 5 should be reviewed.

There is lots of flexibility in the way the text can be studied and/or approached. We trust the various readers will find their way through the maze and enjoy the material they stop to study or, as we and others have said, will enjoy this tour of Ramanujan’s garden and the flowers they pick there.

5In a course where all nonstandard material is included, the instructor might want to spend some time on the multiplicative properties of the η-function. These could be taken from Knopp’s book [16].

We have included descriptions of many special cases and summarized the results of many calculations. Most of the nontrivial calculations used the symbolic manipulation programs MATHEMATICA and/or MAPLE. In order to excite the reader about the flowers in the beautiful garden we are cultivating here, we start each chapter with (what we regard as) a handsome example of what will follow. We have included a number of accessible exercises and research level problems. The latter may be quite challenging and are at times only conjectures. The reader should also approach the many special cases we have included as challenges to obtain independent solutions. They are presented in the spirit of exercises, with solutions supplied by the authors.

Numbering systems. The book consists of seven chapters and a set of bibliographical notes that will be maintained and updated on the web. Chapters are subdivided into sections; these into subsections. Definitions, lemmas, propositions, theorems, exercises, problems and remarks are labeled consecutively as a single group within each section. A typical item is Theorem section.number; number starts with 1 for the first item in the section. Thus, for example, in Chapter 2, Definition 2.32 (in section 2) in our numbering scheme is followed by Lemma 4.1 (in section 4). Equations that will be referenced subsequently in the text are labeled by a decimal: chapter.number; number starts with 1 for the first numbered equation in the chapter. Tables and figures are numbered consecutively in the book.

Acknowledgments. We thank

1. the mathematicians who have had the patience to listen to our lectures on the subject of this book: whether our students at the Hebrew University in Jerusalem or the State University of New York in Stony Brook, who at times have provided us with constructive criticism, or colleagues at various institutions who have invited us to speak about this subject;

2. our graduate students who read previous drafts of the book and pointed out errors and ambiguities (needless to say, the remaining errors and ambiguities are the authors’ responsibilities);

3. our undergraduate students who helped with the MATHEMATICA calculations and the figures;

4. the editorial and technical staff of the American Mathematical Society, especially Deborah Smith and Janet Pecorelli who guided us through the final preparation of the manuscript; and most importantly,

5. our respective wives, who patiently suffered with us through this long project.

Finally we wish to acknowledge and thank the U.S. - Israel Binational Science Foundation, the U.S. National Science Foundation, The Hebrew University and the State University of New York at Stony Brook for their support over the years that helped produce the research for this book; for providing us with the needed time, facilities and equipment; and for making possible visits by each of us to the institution of the other, not only to work on this volume.

Hershel M. Farkas Jerusalem, Israel

Irwin Kra

Stony Brook, NY, U.S.A.

Chapter 1

The modular group and elliptic function theory

This chapter contains mostly well known material on the modular group PSL(2, ℤ) and elliptic curves along with some relatively new material on once punctured tori. We present in §4 a more or less classical treatment of function theory on elliptic curves (a euclidean theory) as a model for subsequent work. It is followed by an outline of the equivalent (noneuclidean) theory on once punctured tori using Poincaré series to illustrate another approach to some of the problems we will encounter in this book. The most important fact for our subsequent chapters from this material is that the sum of the residues of an elliptic function is zero. We introduce and describe the geometry of several families of subgroups of the modular group that will be needed in the sequel. This section (§6) establishes much of the notation that will be used throughout our presentation. The chapter ends with the elementary application to primality testing discussed in the introduction.

Most of the material in this chapter can be found in the references supplied. Much of the material is presented as background, a guide to different approaches, and to establish notation. As pointed out above, an exception is made in subsections of §4, where complete arguments are provided.

1. Möbius transformations

The material in this book deals mostly with compact Riemann surfaces and subgroups of PSL(2, ℤ). We start with a discussion of the more general group: PSL(2, ℂ). Good additional references for this introductory material are [4, Ch. 4], [12, Ch. I] and [21, Ch. I].

An element A = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ PSL(2,ℂ) acts on the Riemann sphere

ℂ̂ = ℂ ∪ {∞} as the Möbius transformation

z ↦ \frac{az + b}{cz + d}.

We think of PSL(2,ℂ) as a topological group with the following topology. Consider SL(2,ℂ) as the closed subset of complex euclidean 4-space defined by the equation

(a, b, c, d) ∈ ℂ^{4}; ad − bc = 1,

and give PSL(2, ℂ) = SL(2, ℂ)/{±I} the quotient topology.

It is convenient to introduce a most important and useful classification

for Möbius transformations. Let A = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ PSL(2,ℂ), A ≠ I. Then

(recall that ad − bc = 1)

A is parabolic ⟺ tr^{2}(A) = (a + d)^{2} = 4,
A is elliptic ⟺ 0 ≤ tr^{2}(A) < 4,
A is hyperbolic ⟺ tr^{2}(A) > 4,

and

A is loxodromic ⟺ tr^{2}(A) ∉ [0, 4].

We note that in our definition hyperbolic is a special case of loxodromic. The (square of the) trace of an element of PSL(2,ℂ) is a conjugacy class invariant; that is,

tr(A) = tr(B ∘ A ∘ B^{−1}) for all A, B ∈ SL(2, ℂ).

Conversely, if A ≠ I and B ≠ I are two nontrivial Möbius transformations^{6} with the same squared trace, then they are conjugate in PSL(2,ℂ); if the two motions are in PSL(2, ℝ) and they have the same squared trace, then we can only conclude that A is conjugate in PSL(2, ℝ) to either B or B^{−1}.

The motion A has either one or two fixed points α, β in ℂ̂; they are given by the formulae

\{α, β\} = \{\frac{1}{2c}((a − d) ± \sqrt{tr^{2}(A) − 4})\} if c ≠ 0,

6We denote the identity matrix and the identity Möbius transformation by the same symbol I.

and

\{α, β\} = \{∞, −\frac{b}{a − a^{−1}} \} if c = 0.

It follows from the above formulae^{7} that A is parabolic if and only if A has one fixed point in ℂ̂, if and only if A is conjugate in PSL(2,ℂ) to

[ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ]. In all other cases A ≠ I has two fixed points and is hence conju-

gate in PSL(2,ℂ) to [ \begin{matrix}κ & 0 \\ 0 & κ^{−1} \end{matrix} ] with κ ∈ ℂ − \{0, 1, −1\}. In this case A is

elliptic if and only if |κ| = 1, A is hyperbolic if and only if κ ∈ ℝ, and A is loxodromic if and only if |κ| ≠ 1.

The square of κ, κ^{2}(A) is called the multiplier of A. The multiplier is defined uniquely except that a number and its reciprocal are the two multipliers of the same Möbius transformation.^{8} For A parabolic or the identity, we set κ^{2}(A) = 1. We note that

κ^{2}(A^{n}) = (κ^{2}(A))^{n}, for all A ∈ PSL(2, ℂ).

Hence only elliptic elements can have finite order. A Möbius transformation is of finite order if and only if its multiplier is a root of unity. The order n of such a motion A is the smallest positive integer n such that κ^{2n}(A) = 1.

The trace and the square root of the multiplier are related by the formula

tr = κ + κ^{−1}.

A parabolic motion A with finite fixed point a and translation length α can be written in normal form as a motion of the sphere as

\frac{1}{A(z) − a} = \frac{1}{z − a} + α

(with α ∈ ℂ^{∗}), or equivalently as an element of PSL(2,ℂ) as

A = [ \begin{matrix}1 + aα & −a^{2}α \\ α & 1 − aα \end{matrix} ] .

The case of infinity as a fixed point corresponds to a = ∞ and −a^{2}α finite and nonzero (this latter quantity is the translation length in this case); thus α = 0 = aα. We record for future use

A′(a) = 1 and A′′(a) = −2α.

7Many times hereafter we will derive onlyformulae for finite values of the parameters (or variables) (here fixed points). We will usually leave it to the reader to make the necessary adjustments for the parameter assuming the value ∞.

8For loxodromic motions, it is always possible to choose the multiplier so that its magnitude is > 1. For primitive elliptic motions we may choose the multiplier so that its argument lies in (0, π].

A motion A with fixed points a and b and multiplier κ^{2} is given in normal form by

\frac{A(z) − a}{A(z) − b} = κ^{2} \frac{z − a}{z − b},

and equivalently as a matrix (of determinant 1)

A = \frac{1}{κ(a − b)} [ \begin{matrix}a − κ^{2}b & ab(κ^{2} − 1) \\ 1 − κ^{2} & κ^{2}a − b \end{matrix} ] .

Simple calculations show that

A′(a) = κ^{2} and A′(b) = κ^{−2}.

It follows that, unless A has order two, the selection of a multiplier allows us to distinguish one fixed point from the other. For loxodromic A, we call a the attractive (respectively, repulsive) fixed point provided |A′(a)| < 1 (respectively, |A′(a)| > 1).

As an example, we consider a sequence of nonparabolic motions

A_{n} = [ \begin{matrix}a_{n} & b_{n} \\ c_{n} & d_{n} \end{matrix} ] ∈ PSL(2,ℂ)

whose elements do not fix ∞ and that converges to a parabolic motion with fixed point α. This means that (we may assume after multiplying the entries of A_{n} by −1, if necessary)

lim_{n→∞} \frac{a_{n} − d_{n}}{2c_{n}} = α and lim_{n→∞} a_{n} + d_{n} = 2.

If, on the other hand, A_{n} fixes ∞ and α_{n} and α = ∞, then

A_{n} = [ \begin{matrix}κ_{n} & α_{n}(κ_{n}^{−1} − κ_{n}) \\ 0 & κ_{n}^{−1} \end{matrix} ]

with

lim_{n→∞} κ_{n} = 1, lim_{n→∞} α_{n} = ∞ and lim_{n→∞} α_{n}(κ_{n}^{−1} − κ_{n}) = b ∈ ℂ^{∗}.

2. Riemann surfaces

More on the material in this section can be found in [6, Ch. I].

Definition 2.1. A Riemann surface M is a one (complex) dimensional connected complex analytic manifold.

This means that we can find open sets {U_{α}; α ∈ A} that cover M (that is, M ⊂ ∪_{α∈A}U_{α}) and homeomorphisms called (local) coordinates

z_{α} : U_{α} → ℂ

whose transition functions

f_{αβ} = z_{α} ∘ z_{β}^{−1} : z_{β}(U_{α} ∩ U_{β}) → z_{α}(U_{α} ∩ U_{β})

are complex analytic.

Definition 2.2. A compact Riemann surface is closed; a noncompact, open.

Definition 2.3. A continuous map

f : M → N

between Riemann surfaces is holomorphic if for every local coordinate z on M and every local coordinate ζ on N, the map ζ ∘ f ∘ z^{−1} is holomorphic. A holomorphic equivalence (an injective holomorphic map of one surface onto another) is a conformal map.

Theorem 2.4. Let f : M → N be a holomorphic map between Riemann surfaces. If M is compact, then f is either constant or surjective.

Proof. If f is not constant, then it is an open mapping. Thus f(M) is open in N. It is closed in N because it is a compact subset of N. Since N is connected f(M) = N.

Definition 2.5. A holomorphic map of a Riemann surface M into the Riemann sphere ℂ̂ other than the constant map with value ∞ is called a meromorphic function. The set of meromorphic functions on M forms a field, K(M).

3. Kleinian groups

See [4, Ch. 5] and [21, Ch. II] for more details on this subject. We outline several aspects of the theory of Kleinian groups. The reader is warned that NOT all claims made are easily verifiable. We will, however, subsequently prove all that we need for our work and is not easily available in other books.

3.1. Generalities. Let X be a subset of the sphere. Consider an inclusion of groups G ⊂ Γ ⊂ PSL(2, ℂ). We say that X is precisely invariant under G in Γ provided

g(X) = X for all g ∈ G and γ(X) ∩ X = ∅ for all γ ∈ Γ − G.

We say that Γ acts discontinuously at the point z in ℂ̂ provided the stabilizer Γ_{z} of z in Γ is finite and there exists a neighborhood U of z that is precisely invariant under Γ_{z} in Γ. The region of discontinuity or ordinary set Ω = Ω(Γ) is the largest (automatically open) set on which the group Γ acts discontinuously. Its complement Λ = Λ(Γ) is the limit set of Γ. We call Γ Kleinian provided the region of discontinuity is nonempty. In this case, the limit set contains zero, one, two or infinitely many points. If Λ contains more than two points, then it is a closed perfect subset of ℂ̂. If Γ is infinite and Kleinian, then the limit set is the set of accumulation points of the orbit under Γ of a single ordinary point of the group.

A group Γ of Möbius transformations is called elementary provided every two elements of infinite order in the group have a common fixed point. If Γ is Kleinian, then it is elementary if and only if Λ(Γ) has two or fewer points. For nonelementary Kleinian groups Γ, the limit set can also be characterized as the smallest nonempty Γ-invariant closed subset of the sphere. The elementary discrete groups are all Kleinian and well known.

Remark 3.1. If Γ is a subgroup of PSL(2,ℂ) and Γ acts discontinuously somewhere on ℂ̂, then Γ is automatically discrete (in the topology for the group PSL(2,ℂ) described above). Hence every Kleinian group is countable. Discreteness does not imply discontinuity. The Picard group, PSL(2, ℤ[ı]), is discrete but does not act discontinuously anywhere on the Riemann sphere. The situation with the Picard group is in marked contrast to the case of PSL(2, ℝ) where discreteness is equivalent to the notion of acting discontinuously on the upper half plane. The reader should not be perplexed by this since even in the case of PSL(2, ℝ), the action on the real line is not discontinuous. Even though we are viewing PSL(2,ℂ) as acting on the extended complex plane, there is another action on the upper half space. The complex plane is the boundary of that space just as the real line is the boundary of the upper half plane.

Definition 3.2. Let G be a group of self-homeomorphisms of a topological space X. The group introduces an equivalence relation on the space. Two points x and y in X are equivalent (modulo G) if there exists a g ∈ G with g(x) = y. The set of equivalence classes is the quotient space X/G. Let π : X → X/G be the quotient map that assigns to each x ∈ X its equivalence class modulo G. We topologize X/G by giving the factor space the weakest topology so that π is continuous.

The factor space Ω/Γ is a union of (at most countably many) Riemann surfaces whenever the group Γ is Kleinian, as a direct consequence of the next exercise.

Exercise 3.3. If Γ is a Kleinian group, then for each z ∈ Ω(Γ), the stabilizer

Γ_{z} = {γ ∈ Γ; γ(z) = z}

is a finite cyclic group. Describe the stabilizer of a limit point of a Kleinian group.

Exercise 3.4. If Γ is a Kleinian group and A ∈ PSL(2,ℂ), then the conjugate group

AΓA^{−1} = {A ∘ γ ∘ A^{−1}; γ ∈ Γ}

is also Kleinian. Further

(a) Ω(AΓA^{−1}) = A(Ω(Γ)), (b) Λ(AΓA^{−1}) = A(Λ(Γ)) and

(c) Ω(AΓA^{−1})/AΓA^{−1} ≅ Ω(Γ)/Γ.

Definition 3.5. Let G be a group of Möbius transformations operating on a set D ⊂ ℂ̂. A fundamental domain for the action of G on D is a connected open set ω = ω(G) bounded by analytic arcs ∪_{i∈I}(c_{i}^{+} ∪ c_{i}^{−}) satisfying:

1. For each x ∈ D, there exists a z ∈ cl ω and a g ∈ G such that g(x) = z.

2. If z_{1} and z_{2} belong to ω and g(z_{1}) = z_{2} for some g ∈ G, then g is the identity.

3. For each i ∈ I, there exists a g_{i} ∈ G with g_{i}(c_{i}^{+}) = c_{i}^{−}.

A fundamental set for the action of G on D is a set containing exactly one point for each G-orbit.

Roughly speaking, Ω/Γ is obtained by boundary identifications on the closure of a fundamental domain for the action of Γ on Ω. We let

Δ_{1}, Δ_{2}, ..., Δ_{i}, ...

be a maximal set of Γ-inequivalent components of Ω, and we let

Γ_{i} = {γ ∈ Γ; γ(Δ_{i}) = Δ_{i}}

be the stabilizer of Δ_{i} in Γ. The stabilizer subgroups are finitely generated whenever the group itself is finitely generated. But, this is far from obvious. It is quite obvious, however, that

Ω/Γ ≅ ⋃_{i} Δ_{i}/Γ_{i}.

It turns out that the above is a finite (disjoint) union whenever Γ is finitely generated. As remarked earlier, it is easy to conclude that we have at most a countable union of surfaces.

A Kleinian group Γ is called a function group provided it has an invariant component,^{9} that is, provided there is a component Δ of Ω with γ(Δ) = Δ for all γ ∈ Γ. A finitely generated Kleinian group can have at most two invariant components.^{10} The stabilizers of the components of the ordinary set of a Kleinian group are, of course, function groups. A function group is called a b-group (b after boundary or Bers) provided its invariant component is simply connected. A Kleinian group is called Fuchsian if it is conjugate to a subgroup of PSL(2,ℝ). A b-group is Fuchsian if and only if its invariant component is a disc.

A Kleinian group Γ that leaves invariant a disc Δ is automatically Fuchsian with Δ contained in Ω(Γ). The limit set is, in this case, a subset of ∂(Δ), the set theoretic boundary of the domain Δ. The group is called of the first kind if Λ = ∂(Δ) and of the second kind otherwise. A Fuchsian group of the first kind Γ represents on Ω = Ω(Γ) a Riemann surface and its mirror image, while a group of the second kind represents a single Riemann surface Ω/Γ: the double of Δ/Γ. In this latter case, the surface Ω/Γ is Δ/Γ glued to its mirror image along their common boundary, (∂(Δ) − Λ(Γ))/Γ.

9By abuse of language, components of Ω(G) are also called components of Γ.

10As with many claims in this section, this is not obvious but easily proven. We will not need this fact for our presentation.

3.2. The situation of interest. We consider the special case that we study in this book: a Fuchsian group G operating on a disc Δ and the natural projection P : Δ → Δ/G. The map P is locally one-to-one at each z_{o} ∈ Δ whose stabilizer G_{z_{o}} in G is trivial; if μ = |G_{z_{o}}|, it is locally μ-to-one at z_{o}. In the latter case, z_{o} is called a branch point of order μ (it is also called a ramification point of order μ − 1). The point P (z_{o}) ∈ Δ/G is called a branch value of the map P. A puncture x on Δ/G is a region D^{∗} ⊂ Δ/G that is conformally equivalent to 𝔻^{∗} = {0 < |z| < 1} (say f : 𝔻^{∗} → D^{∗} is this equivalence) with the property that P is unramified over D^{∗} and for all sequences {z_{n}} in 𝔻^{∗} with lim_{n} z_{n} = 0, the sequence {P (z_{n})} ⊂ D^{∗} is discrete in Δ/G (this sequence converges to the puncture x). We assign the branch number ∞ to the puncture x. A puncture on Δ/G is the image of a cusp z on the boundary of Δ for the group G. If we define Δ̄ to be the union of Δ and the cusps of G (these are the fixed points of the parabolic elements of G), then P extends by continuity to Δ̄ and P (Δ̄) = \overline{Δ/G} is the completion^{11} of Δ/G obtained by filling in the punctures. For the puncture x on Δ/G, we can choose a horodisc D precisely invariant under a cyclic parabolic subgroup H ⊂ G with P (D) = D^{∗} (the fixed point z of the parabolic elements in H is on the boundary of D and P (z) = x).

The group G and the surfaces Δ/G and \overline{Δ/G} are of finite (analytic) type (p, n) if \overline{Δ/G} is a compact surface of genus p and Δ/G has a total of n punctures and branch values. If {z_{1}, z_{2}, ..., z_{n}} is the set of points in Δ̄ that project to the punctures and branch values, and if μ_{j} is the order of the branch point or cusp z_{j}, then the n + 2-tuple

(p, n; μ_{1}, μ_{2}, ..., μ_{n})

is the signature of the group G and the orbifolds Δ/G and \overline{Δ/G}; their (negative) Euler characteristic is

χ(G) = 2p − 2 + ∑_{j=1}^{∞} (1 − \frac{1}{μ_{j}} ) ,

where we use the usual convention \frac{1}{∞} = 0.

11In many of the interesting cases, the compactification.

4. The elliptic paradise

The special case of surfaces of genus one^{12} is much better understood than the more general case of higher genus. We discuss in this section a classical theory. It is a model for the subsequent work. This is one of the very few cases where complete computations are possible.

4.1. The family of tori. More on material in this section can be found in [1, Ch. 7] and [14, §1.2].

We begin by constructing a (one could almost say “the”) family of tori. This procedure is well known. It is nevertheless a good starting point. Let τ be any point in the upper half plane

ℍ^{2} = {τ ∈ ℂ; ℑτ > 0}.

The point τ determines a free abelian group G̃ = G̃_{τ} generated by the two translations

Ã(z) = z + 1, B̃(z) = z + τ.

The complex plane ℂ factored by the equivalence relation defined by G̃_{τ} is a torus that we will denote by the symbol T_{τ}:

T_{τ} = ℂ/G̃_{τ} .

Let a ∈ ℂ be arbitrary. A fundamental domain ω̃ = ω(G̃_{τ}) for the action of the group G̃ on ℂ is the interior of the parallelogram P_{a} with vertices a, a + 1, a + 1 + τ, a + τ; the interior of P_{a} with the edges from a + τ to a to a + 1, excluding the vertices a + τ and a + 1, forms a fundamental set for this action.

The choice of a is essentially immaterial, but, as will be seen, an appropriate choice will simplify proofs. Abel’s theorem^{13} tells us that every torus^{14} is of the above form (that is, the quotient of the complex plane by a group of translations). As a consequence, the upper half plane can be naturally identified with the Teichmüller space T(1, 0) of compact surfaces of genus 1. A point in this Teichmüller space is a “marked torus”; the marking is provided by the point τ which gives an isomorphism from the fundamental group of the torus to the covering group G̃.

12To the best of our knowledge the catchy title that we assigned to this section is due to Lipman Bers.

13Not proven in this book. This special uniformization theorem for tori is also a fairly trivial consequence of the general uniformization theorem (that there are only three distinct simply connected Riemann surfaces).

14For our purposes, we can define a torus as a surface conformally equivalent to T_{τ} with τ ∈ ℍ^{2}. A more abstract (equivalent, of course) definition of a torus is a Riemann surface whose fundamental group is isomorphic to ℤ ⊕ ℤ.

Figure 1. Fundamental polygon. P_{0}

We should point out that we have, in reality, two definitions of the torus. The one we have emphasized so far is a manifold ℂ factored by the action of a discontinuous group G̃; hence T_{τ} is a complex analytic manifold. The lattice points L of the group G̃ are the points

L = L(G̃) = {n + mτ; n and m ∈ ℤ}.

The lattice points form a (normal) subgroup of the abelian group of additive complex numbers. Hence T_{τ} = ℂ/L is also a commutative group, in fact a one dimensional complex Lie group.

Theorem 4.1. Two points τ_{1}, τ_{2} ∈ ℍ^{2} determine conformally equivalent tori if and only if there exists a Möbius transformation M ∈ PSL(2, ℤ) with τ_{2} = M(τ_{1}).

Proof. Assume that f is a conformal map from T_{τ_{1}} onto T_{τ_{2}}. Then there exists a conformal selfmap F of ℂ so that

ℂ ⟶^{F} ℂ
↓  ↓
T_{τ_{1}} ⟶^{f} T_{τ_{2}}

commutes, where the vertical arrows are the natural projections. Since F is conformal, it is of the form

z ↦ az + b, a ∈ ℂ^{∗} = ℂ − {0}, b ∈ ℂ.

Translation by b induces a conformal selfmap of T_{τ_{2}}. Hence we may and do assume that b = 0. The affine map F must conjugate the group G̃_{τ_{1}} to G̃_{τ_{2}} (that is, F G̃_{τ_{1}}F^{−1} = G̃_{τ_{2}}). Let us call this conjugation θ. Identifying the translation z ↦ z + b with the complex number b, we see that the θ(1) = F (1) = a and θ(τ_{1}) = F (τ_{1}) = aτ_{1} must be generators for G̃_{τ_{2}}. Thus

there exists a matrix A = [ \begin{matrix}α & β \\ γ & δ \end{matrix} ] ∈ GL(2, ℤ) so that

a = α + βτ_{2} and aτ_{1} = γ + δτ_{2}.

Hence

τ_{1} = \frac{γ + δτ_{2}}{α + βτ_{2}} .

Since both τ_{1} and τ_{2} belong to ℍ^{2}, A must belong to SL(2, ℤ), and M =

[ \begin{matrix}α & −γ \\ −β & δ \end{matrix} ].

Conversely, if τ_{2} = M(τ_{1}) with M as above, then we define

F (z) = (α + βτ_{2})z, z ∈ ℂ.

Simple calculations show that

F (z + 1) = F (z) + (α + βτ_{2})

and

F (z + τ_{1}) = F (z) + (γ + δτ_{2}), for all z ∈ ℂ.

It follows that F defines a conformal map of T_{τ_{1}} onto T_{τ_{2}}.

Exercise 4.2. Let G be a group of Möbius transformations. Assume that G is discontinuous and consists in addition to the identity of only parabolic transformations. Show that G is either cyclic and hence conjugate to the group generated by the selfmap^{15} B of ℂ̂

B : z ↦ z + 1, z ∈ ℂ̂,

or G is conjugate to G̃_{τ} for some τ ∈ ℍ^{2}.

Theorem 4.3. A fundamental domain for the action of Γ =PSL(2, ℤ) on ℍ^{2} is the open region ω(PSL(2, ℤ)) in the upper half plane bounded by the lines ℜτ = −\frac{1}{2}, ℜτ = \frac{1}{2} and the semi-circle |τ| = 1. A fundamental set for this action can be obtained by adjoining the circular arc from ı to \frac{1}{2} + \frac{\sqrt{3}}{2}ı (including both end points) and the vertical line segment from \frac{1}{2} + \frac{\sqrt{3}}{2}ı to ı∞.

15We will reserve the symbol B for this motion in most of the rest of this manuscript, in particular, when dealing with subgroups of the modular group. In our discussion of tori we use the symbol Ã for this motion.

Figure 2. Fundamental domain for the modular group Γ.

Proof. Let us denote by ω the set described by the last sentence in the statement of the theorem. Assume that we have two points τ_{1} and τ_{2} in ω that are equivalent under Γ. Without loss of generality we may assume that

ℑτ_{2} ≥ ℑτ_{1}.

Let

τ_{2} = \frac{aτ_{1} + b}{cτ_{1} + d} with C = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ SL(2, ℤ).

Since

ℑτ_{2} = \frac{ℑτ_{1}}{|cτ_{1} + d|^{2}} ,

we must have that |cτ_{1} +d| ≤ 1. We now exploit the fact that we are dealing with integer matrices.

If c = 0, then ad = 1 and both a and d must be ±1. Without loss of generality we may assume that a = 1 = d. It follows that τ_{2} = τ_{1} + b. Hence |b| = |ℜτ_{2} − ℜτ_{1}| < 1 and b = 0.

Assume now that c ≠ 0. The condition |τ_{1} + \frac{d}{c}| ≤ \frac{1}{|c|} implies that |c| = 1. For if |c| ≥ 2, then τ_{1} would be at a distance at most \frac{1}{2} from ℝ, which is impossible since the point in ω nearest to ℝ is \frac{1}{2} + \frac{\sqrt{3}}{2}ı. Without loss of generality we may now assume that c = 1 and hence |τ_{1} + d| ≤ 1. The geometric shape of ω tells us that this inequality is possible only for |d| ≤ 1. The inequality |τ_{1} + 1| ≤ 1 is never satisfied for points τ_{1} ∈ ω since exp \frac{2πı}{3} ∉ ω. The inequality |τ_{1} − 1| ≤ 1 is satisfied only by the point exp \frac{πı}{3} ∈ ω. For this point |cτ_{1} + d| = 1. Hence ℑτ_{2} = ℑτ_{1}, and by the shape of ω, τ_{2} = τ_{1}. There remains only the case c = 1 and d = 0. Hence b = −1. Since |τ_{1}| ≤ 1 and |τ_{1}| ≥ 1, we conclude that |τ_{1}| = 1 and τ_{2} = a− \frac{1}{τ_{1}} = a−τ̄_{1}. Hence ℜ(τ_{1} + τ_{2}) = a, and this is possible only when a = 0. Thus τ_{2} = −\frac{1}{τ_{1}} , which tells us that τ_{1} = ı = τ_{2}.

It remains to show that every τ ∈ ℍ^{2} is Γ-equivalent to at least one point in the closure of ω. It is convenient to show first that the region ω(PSL(2, ℤ)) is the same as

ω_{1} = \{z ∈ ℍ^{2}; |ℜz| < \frac{1}{2} and |cz + d| > 1 for all (c, d) ∈ ℤ^{2} − \{(0, 0)\}\} .

It is obvious that ω_{1} ⊂ ω(PSL(2, ℤ)). For the reverse inclusion, if z ∈ ω(PSL(2, ℤ)), and (c, d) ∈ ℤ^{2} − {(0, 0)}, then

|cz + d|^{2} = (cℜz + d)^{2} + (cℑz)^{2} =
(c^{2})(ℜz^{2} + ℑz^{2}) + 2cdℜz + d^{2} > c^{2} − cd + d^{2} ≥ 1.

Every τ ∈ ℍ^{2} is certainly equivalent to a τ_{1} ∈ ℍ^{2} with maximal height^{16} (that is, to a τ_{1} ∈ ℍ^{2} with |cτ_{1} +d| ≥ 1, for all integers (c, d) ∈ ℤ^{2} −{(0, 0)}). Translating by powers of B we can certainly get an equivalent point to land on the closure of ω_{1} (since the height of a point is unaffected by the translation).

A point τ in the interior of this fundamental domain ω(PSL(2, ℤ)) represents a very particular kind of marking on the surface. The condition that τ is in (the interior of) the fundamental domain can be restated as

|τ| > 1 and |τ| < |τ ± 1|.

Thus on the torus T_{τ} we are choosing a basis for the fundamental group with Ã representing the shortest closed geodesic^{17} and B̃ representing the shortest curve that is not homotopic to a power of Ã. Almost all tori (except those corresponding to boundary points of the fundamental domain) have a unique marking of this type. The factor space ℍ^{2}/PSL(2, ℤ) is the Riemann or moduli space of surfaces of genus 1, R(1, 0). It is the complex sphere ℂ∪{∞} with three distinguished points, a Riemann surface (orbifold) of signature (0, 3; 2, 3, ∞). The distinguished points correspond to the fixed points τ =

16If c ≠ 0, then C maps the upper half plane {ℑτ ≥ 1} to the closed disc in ℍ^{2}of diameter \frac{1}{c^{2}} tangent to ℝ at \frac{a}{c} .

17In any flat (zero curvature) metric, for example, |dz|.

ı and τ = \frac{1+\sqrt{3}ı}{2} of the elliptic motions [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] and [ \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ] of order

2 and 3 (respectively) and the fixed point τ = ∞ of the parabolic motion

[ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ] . We will see later that these fixed points determine special tori.

4.2. The algebraic curve associated to a torus. The material on tori in this section is included in most books on basic function theory, for example, in [1, Ch. 7]. The material on abstract surfaces comes from [6, Chs. I and III]. Fix a point τ ∈ ℍ^{2}.

Definition 4.4. A meromorphic function f on ℂ is elliptic or (doubly) periodic (with periods 1 and τ) if

f(z + 1) = f(z) = f(z + τ), for all z ∈ ℂ.

It is clear that the space of periodic functions can be naturally identified with K(T_{τ}). If f ≠ 0 is a meromorphic function on a Riemann surface X and z is a local coordinate on X vanishing at x, then f has a Laurent series expansion

f(z) = ∑_{i=n}^{∞} a_{i}z^{i}, a_{n} ≠ 0.

We call n the order of f at x, in symbols, ord_{x}f. If the periodic function f assumes the value c at z ∈ ℂ, then it assumes the same value c at every point in ℂ that is G̃_{τ} -equivalent to z. Hence in counting the number of solutions x (usually with multiplicity) of f(x) = c, we include only inequivalent points x. We are, in effect, counting solutions on the torus T_{τ}. A periodic function f also has a well defined residue, res_{x}f, at each point x ∈ ℂ. Again, the concept of residues should be viewed on the torus (since f has the same residue at G̃_{τ} -equivalent points).

Remark 4.5. Let X be a Riemann surface and 0 ≠ f ∈ K(X). It makes sense to define ord_{x}f for each x ∈ X. However, the residue of a function is not a well defined concept on an arbitrary Riemann surface. On an arbitrary surface one can define meromorphic one forms φ. These are assignments of a meromorphic function f to each local coordinate z on the surface such that f(z)dz is invariant under coordinate changes. It then makes sense to define res_{x}φ. There is a canonical isomorphism on every torus between the space of meromorphic functions and the space of meromorphic one forms, because of the presence of a regular (holomorphic) global one form without zeros (namely, the projection of dz to T_{τ}). For each q ∈ ℤ, one defines a meromorphic q-form as an assignment of a meromorphic function φ to each local coordinate z such that

φ(z)dz^{q}

is invariant under coordinate changes. These spaces are useful tools in the study of function theory on surfaces.

Theorem 4.6. The sum of the residues (over a maximal set of inequivalent points) of an elliptic function is 0.

Proof. Let f be a periodic function. Since the poles of a meromorphic function are isolated, we can choose a ∈ ℂ so that f is regular on the boundary ∂P_{a} of the period parallelogram P_{a}. It follows that

∑_{x∈P_{a}} res_{x}f = \frac{1}{2πı} ∫_{∂P_{a}} f(z)dz

= \frac{1}{2πı} (∫_{a}^{a+1} f(z)dz + ∫_{a+1}^{a+1+τ} f(z)dz + ∫_{a+1+τ}^{a+τ} f(z)dz + ∫_{a+τ}^{a} f(z)dz) .

Using the periodicity of the function f and a change of variable, we conclude the sum of the first and third integrals, as well as the sum of the second and fourth integrals, on the last line must be zero.

Corollary 4.7. Let f be a nonconstant elliptic function and let c ∈ ℂ̂. The number of solutions of f(z) = c is independent of c and will be called the degree, deg f, of the function f.

Proof. Assume that c is finite. Then g = \frac{f′}{f−c} ≠ 0 is an elliptic function. Since res_{x}g = ord_{x}(f − c), the corollary for f follows from the theorem for g.

Corollary 4.8. A nonconstant elliptic function must have poles.

Remark 4.9. The genus p = p(X) of a compact Riemann surface X is defined as \frac{1}{2}rank H_{1}(X, ℤ). It is a complete topological invariant for compact surfaces: two compact Riemann surfaces are homeomorphic if and only if they have the same genus. Let f : X → Y be a nonconstant map between compact Riemann surfaces. We have already seen that f is surjective. Let x ∈ X. We can find local coordinates z on X vanishing at x and ζ on Y = f(X) vanishing at f(x) such that in terms of these coordinates

ζ = z^{b_{f} (x)+1},

where b_{f}(x) is a nonnegative integer (which equals zero at all but finitely many x ∈ X) known as the branch number of f at x. We define the degree of f, deg f, as

∑_{x∈f^{−1}(y)}(b_{f}(x) + 1); y ∈ Y.

It is independent of the choice of y, and the Riemann Hurwitz relation

2p(X) − 2 = deg f (2p(Y) − 2) + ∑_{x∈X} b_{f}(x)

holds.

Corollary 4.10. A torus does not carry any meromorphic functions of degree one.

Proof. (Topological) Such a function would establish an isomorphism (in particular, a topological equivalence) between the torus (whose fundamental group is ℤ^{2}) and the Riemann sphere (which is simply connected). (Analytic) The sum of the residues of such a function would have to be nonzero.

Theorem 4.11. Let f be a nonconstant elliptic function of degree n. Let a_{1}, ..., a_{n} be the zeros of f (listed according to their multiplicities) and b_{1}, ..., b_{n} its poles. Then

∑_{i=1}^{n}(a_{i} − b_{i}) = 0

(in the group T_{τ}).

Proof. By properly choosing a ∈ ℂ (close to 0), we may assume that f is regular and nonzero on the boundary of P_{a}. By the residue theorem, the sum to be evaluated is

\frac{1}{2πı} ∫_{∂P_{a}} z \frac{f′(z)}{f(z)}dz

= \frac{1}{2πı} (∫_{a}^{a+1} + ∫_{a+1}^{a+1+τ} − ∫_{a+τ}^{a+τ+1} − ∫_{a}^{a+τ} ) z \frac{f′(z)}{f(z)}dz.

We now group the first and third integral, the second and fourth integral, and use the periodicity of the elliptic function \frac{f′}{f} to continue the evaluation as

(\frac{−τ}{2πı} ∫_{a}^{a+1} + ∫_{a}^{a+τ} ) \frac{f′(z)}{f(z)}dz

= \frac{1}{2πı} (−τ(log f(a + 1) − log f(a)) + (log f(a + τ) − log f(a))) = n+mτ,

for some integers n and m.

Remark 4.12. There is a converse to the above theorem. It will be established in Chapter 2; see Theorem 5.1 of that chapter. The theorem and its converse are part of a more general theory. On an arbitrary compact Riemann surface X of genus p > 0, one constructs a canonical homology basis (for H_{1}(X, ℤ))

a_{1}, ..., a_{p}, b_{1}, ..., b_{p}

(this means that the intersection numbers of the cycles (curves) satisfy

a_{i} × a_{j} = 0 = b_{i} × b_{j}, a_{i} × b_{j} = δ_{ij}, 1 ≤ i, j ≤ p,

where δ_{ij} is the Kronecker δ-symbol). Then one proves that there exists a dual basis

ω_{1}, ..., ω_{p}

for the space of holomorphic one forms on X. This means that

∫_{a_{i}} ω_{j} = δ_{ij}, i, j = 1, ..., p.

The period matrix π defined by

π_{ij} = ∫_{b_{i}} ω_{j}, i, j = 1, ..., p,

is a symmetric matrix whose imaginary part is positive definite. It follows that the p columns of the p × p identity matrix together with the p columns of π generate a rank 2p lattice L(X) ⊂ ℂ^{p}. The quotient J(X) = ℂ^{p}/L(X) is a p-dimensional torus known as the Jacobi variety of the Riemann surface X. There is a holomorphic mapping

φ : X → J(X)

defined by sending the point x ∈ X to the projection to J(X) of the vector ∫_{x_{o}}^{x} ω, where x_{o} is a fixed point in X (the base point for the embedding) and ω is the vector valued (column) differential on X whose components are ω_{1}, ..., ω_{p}. The map φ is injective and of maximal rank everywhere.

Exercise 4.13. Show that for a surface T_{τ} of genus one, J(T_{τ}) = T_{τ} and φ is the identity map.

Exercise 4.14. The map C : z ↦ \frac{z}{cτ+d} is (induces) a conformal map from T_{τ} to T_{\frac{aτ+b}{cτ+d}}, and hence the map f ↦ f ∘ C is an isomorphism of K(T_{\frac{aτ+b}{cτ+d}}) onto K(T_{τ}).

The divisor group of the surface X is the free (multiplicative) group on the points of X; thus a divisor D is a formal product D = x_{1}^{α_{1}} ... x_{n}^{α_{n}} with n ∈ ℤ^{+} ∪ {0}, α_{i} ∈ ℤ, and x_{i} ∈ X, 1 ≤ i ≤ n. The degree of the divisor D is then ∑_{i=1}^{n} α_{i}. If 0 ≠ f ∈ K(X), then the zeros and poles of f form a principal divisor

(f) = ∏_{x∈X} x^{ord_{x}f}

of degree 0. Since the divisors on X and its Jacobi variety are (abelian) groups, the map φ extends in an obvious way to divisors: the divisor D is sent by this extension φ to ∑_{i=1}^{p} α_{i}φ(x_{i}). The generalization of the last result now reads as follows.

Theorem 4.15 (Abel). A divisor D of degree 0 on the compact Riemann surface X of positive genus is principal if and only if φ(D) = 0.

The theorem has as an immediate consequence: the

Corollary 4.16. If p(X) = 1, then φ : X → J(X) is an isomorphism.

Every torus is also an algebraic curve of a particularly simple form. To obtain the curve associated to a Riemann surface, one must produce “good” meromorphic functions on the surface. In the case under study, we must produce periodic functions. We know that a torus cannot carry a meromorphic function of degree one. Thus we seek functions of degree two. The Riemann Hurwitz formula tells us that such a function must have 4 branch points. It involves no loss of generality to assume that infinity is a branch value (that is, the function we seek should have a single double pole). Finally, the location of the pole is arbitrary since the group of conformal automorphisms of a torus is transitive (the translation z ↦ z + a maps the origin to the equivalence class of the arbitrary point a ∈ ℂ). Thus we may as well place the pole at the origin of the torus. The Laurent series expansion of such a function f about the origin is

(1.1)  f(z) = \frac{a_{−2}}{z^{2}} + a_{0} + a_{1}z + ...

since a_{−1}, the residue of f at the origin, must vanish. We can normalize so that a_{−2} = 1 and a_{0} = 0 by choosing

g(z) = \frac{1}{a_{−2}} (f(z) − a_{0}) = \frac{1}{z^{2}} + A_{1}z + ... .

Since the function h defined by

h(z) = g(z) − g(−z), z ∈ ℂ,

is also elliptic and holomorphic (because the single pole has been killed), it follows that g is an even function. We conclude that if an elliptic function of degree 2 with a single pole at the origin exists, it must be an even function. It remains to construct this function.

We introduce the simplest doubly periodic function, the Weierstrass ℘- function:

℘(z) = \frac{1}{z^{2}} + ∑_{(n,m)∈ℤ⊕ℤ−\{(0,0)\}}\{\frac{1}{(z − n − mτ)^{2}} − \frac{1}{(n + mτ)^{2}} \} , z ∈ ℂ.

Theorem 4.17. The infinite series defining ℘ converges uniformly and absolutely (as a series of meromorphic functions) on compact subsets of ℂ and defines a doubly periodic function (with periods 1 and τ).

Proof. We write ω = n+mτ, with n and m ∈ ℤ. For z ∈ ℂ, with |ω| > 2|z|, we have

|\frac{1}{(z − ω)^{2}} − \frac{1}{ω^{2}} | = |\frac{z(2ω − z)}{ω^{2}(z − ω)^{2}} | ≤ \frac{10|z|}{|ω|^{3}}.

Thus the series defining ℘ converges uniformly and absolutely on compact subsets of ℂ provided

∑_{ω≠0} \frac{1}{|ω|^{3}} < ∞.

We leave the reader the task of showing that this last series converges. It is quite obvious that

\begin{matrix}℘′(z) = & −\frac{2}{z^{3}} − ∑_{(n,m)∈ℤ⊕ℤ−\{(0,0)\}} \frac{2}{(z−n−mτ)^{3}} \\ = & −2 ∑_{(n,m)∈ℤ⊕ℤ} \frac{1}{(z−n−mτ)^{3}} , z ∈ ℂ \end{matrix} ,

is an elliptic function. Thus each of the entire functions

z ↦ ℘(z + 1) − ℘(z) and z ↦ ℘(z + τ) − ℘(z)

is constant. Since ℘ is an even function, evaluation of these last two functions at −\frac{1}{2} and −\frac{τ}{2}, respectively, shows that the constants are 0.

It is quite easy to show that (a formal calculation will do since all the series involved converge)

(1.2) ℘(z) = \frac{1}{z^{2}} + ∑_{k=1}^{∞}(2k + 1) ∑_{(n,m)∈ℤ⊕ℤ−\{(0,0)\}} \frac{1}{(n + mτ)^{2k+2}} z^{2k}, z ∈ ℂ.

It is obvious that

G_{k} = G_{k}(τ) = ∑_{(n,m)∈ℤ⊕ℤ−\{(0,0)\}} \frac{1}{(n + mτ)^{k}}

converges for every integer k ≥ 3. It is also obvious that G_{k} = 0 for all odd k since both n + mτ and −n − mτ appear in the sum. We thus once again see that ℘ is an even function.

The function ℘ has a double pole at the lattice points and is regular elsewhere on the plane. The derivative, ℘′, of the ℘-function is again periodic with a triple pole at the origin.

Theorem 4.18. For every doubly periodic function f (with periods 1 and τ), we can find rational functions a and b so that

f = a(℘)℘′ + b(℘).

Proof. Let z_{1}, ..., z_{k} be a maximal list of G̃-inequivalent points where f has poles. We assume that the equivalence class of the origin has been temporarily removed from this list (if f has a pole there). By choosing sufficiently large integers μ_{1}, ..., μ_{k},

f_{1} = f ∏_{i=1}^{k}(℘ − ℘(z_{i}))^{μ_{i}}

is a doubly periodic function which is holomorphic except at the lattice points. For the positive integer k, ℘^{k} has a pole of order 2k at the origin, and ℘^{k−1}℘′ has a pole of order 2k + 1 at the origin. Thus we can find polynomials c and d such that

f_{2} = f_{1} − c(℘) − d(℘)℘′

is holomorphic except for perhaps a simple pole at the lattice points. If this doubly periodic function had a simple pole at the origin, then it would define a degree 1 function from the torus T_{τ} to the sphere. Since this is impossible for topological reasons, f_{2} is constant. This gives the desired representation for the function f.

The elliptic functions ℘ and ℘′ are not (algebraically) independent. To derive “the” relation between them, we compute the first few terms of the Laurent series expansion about 0 for

℘′(z)^{2} − 4℘(z)^{3} + 60G_{4}℘(z) = −140G_{6} + ... .

The left hand side is an elliptic function. The right hand side shows that this function is free from singularities, hence constant. We conclude that

(1.3)  ℘′(z)^{2} = 4℘(z)^{3} − 60G_{4}℘(z) − 140G_{6}, for all z ∈ ℂ.

We now let y = ℘′(z), x = ℘(z) and rewrite the above equation as

(1.4)  y^{2} = 4(x − e_{1})(x − e_{2})(x − e_{3}),

which is obviously irreducible. Thus the field of meromorphic functions on the torus T_{τ} is isomorphic to the function field

\frac{ℂ(x)[y]}{(y^{2} − 4(x − e_{1})(x − e_{2})(x − e_{3}))}.

It is most important to note that the passage from the covering group of the torus to its algebraic curve involved the transcendental step of constructing the Weierstrass ℘-function. It is also customary to set g_{2} = 60G_{4} and g_{3} = 140G_{6} and translate (1.3) to

℘′(z)^{2} = 4℘(z)^{3} − g_{2}℘(z) − g_{3}.

Remark 4.19. Conversely, every curve determines an algebraic function field in one variable and a covering group (via the general uniformization theorem). The situation in genus 1 is particularly simple since every torus can be defined by essentially the same equation (1.4) with variable parameters (e_{1}, e_{2} and e_{3}).

For an arbitrary compact surface X of genus ≥ 2, one can find two meromorphic functions z and w on the surface that generate the function field K(X). Further, these two functions satisfy an irreducible polynomial identity of the form

∑_{ij} a_{ij}z^{i}w^{j} = 0, a_{ij} ∈ ℂ.

For the special case under consideration, it remains to identify the constants e_{i}, i = 1, 2, 3. The elliptic function ℘′ is odd. Hence

℘′(1 − z) = ℘′(τ − z) = ℘′(1 + τ − z) = −℘′(z).

By setting z = \frac{1}{2}, \frac{τ}{2} and \frac{1+τ}{2} , respectively, we see that ℘′ vanishes at the three half periods. (We have computed the divisor (℘′).) We conclude that we may take

e_{1} = ℘ (\frac{1}{2}) , e_{2} = ℘ (\frac{τ}{2} ) , e_{3} = ℘ (\frac{1 + τ}{2} ) .

We also observe that since ℘ is of degree 2, the three values e_{i} are distinct.

The Weierstrass ℘-function depends on two variables z and τ. It is a meromorphic function of these two variables; its value at (z, τ) ∈ ℂ × ℍ^{2} should be denoted by the symbol ℘(z, τ). The semidirect product of ℤ^{2} and PSL(2, ℤ) acts on the function ℘. We have for all (z, τ) ∈ ℂ × ℍ^{2},

℘(z + n + mτ, τ) = ℘(z, τ) for n, m ∈ ℤ,

and

(1.5)  ℘ (\frac{z}{cτ + d}, \frac{aτ + b}{cτ + d}) = (cτ + d)^{2} ℘(z, τ),

for all γ = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ PSL(2, ℤ). To verify the second formula, let ẑ = \frac{z}{cτ+d}

and τ̂ = \frac{aτ+b}{cτ+d}. Then^{18}

\frac{1}{cτ + d} = a − cτ̂ and \frac{τ}{cτ + d} = −b + dτ̂.

It follows that z ↦ ℘(ẑ, τ̂) defines a function in K(T_{τ}). This function is holomorphic except at the lattice points, where it has a double pole. Since its Laurent series about the origin is

(\frac{z}{cτ + d})^{−2} + a_{2} (\frac{z}{cτ + d})^{2} + ...,

we obtain (1.5) by comparison with (1.1).

It also follows from (1.2) and (1.5) that

\frac{1}{ẑ^{2}} + ∑_{k=1}^{∞}(2k + 1)G_{2k+2}(τ̂)ẑ^{2k} = γ′(τ)^{−1} (\frac{1}{z^{2}} + ∑_{k=1}^{∞}(2k + 1)G_{2k+2}(τ)z^{2k}) .

Comparison of Laurent series coefficients yields

G_{2k+2}(γ(τ))γ′(τ)^{k+1} = G_{2k+2}(τ), for all k ∈ ℤ^{+}, γ ∈ Γ, τ ∈ ℍ^{2}.

Thus, in language to be introduced subsequently (in §1 of Chapter 3), for each positive integer k, G_{2k+2} is an automorphic k + 1-form for the modular group Γ. It has a limit as τ approaches ı∞; as a matter of fact, there is a connection to the Riemann ζ-function

lim _{τ→ı∞}G_{k}(τ) = ∑_{n∈ℤ,n≠0}n^{−k} = \{ \begin{matrix}0 for odd k ≥ 3 \\ 2ζ(k) for even k \end{matrix} .

Hence, using symbols to be introduced in Chapter 3, G_{2k+2} ∈ 𝔸_{k+1}^{+}(ℍ^{2}, Γ). It follows, using the same notation and known values of the Riemann ζfunction (see, for example, [26, pg. 91]), that

G_{4}^{3} − \frac{2ζ(4)^{3}}{ζ(6)^{2}} G_{6}^{2} ∈ 𝔸_{6}(ℍ^{2}, Γ).

z = α + βτ, α, β ∈ ℝ,
ẑ = (αa − βb) + (−αc + βd)τ̂.

18It is useful to record that if we write then

4.3. Invariants for tori. For more information see [1, Ch. 7]. The theory for tori we have discussed so far can be described as the transcendental theory (involving the parameter τ). To begin to study the algebraic theory (parameters λ and ȷ), we proceed to describe the covering of the Riemann space R(1, 0) by the Teichmüller space T(1, 0). First of all, the modular group PSL(2,ℤ) has a big torsion free normal subgroup of finite index, the level 2 principal congruence subgroup

Γ(2) = \{[ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ PSL(2, ℤ); a ≡ d ≡ 1(mod 2), b ≡ c ≡ 0(mod 2)\} .

This group is freely generated by the parabolic motions B^{2} = [ \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ] and

C = [ \begin{matrix}1 & 0 \\ −2 & 1 \end{matrix} ].

Figure 3. Fundamental domain for the level 2 principal congruence subgroup Γ(2).

The groups Γ(2) and PSL(2, ℤ) fit into a short exact sequence

1 → Γ(2) → PSL(2, ℤ) ⟶^{θ} Perm(3) → 1,

where Perm(3) is the permutation group on three letters. The group Γ(2) is the holomorphic universal covering group of the thrice punctured sphere ℂ − {0, 1}. The permutation group Perm(3) acts on the thrice punctured sphere to produce the moduli space R(1, 0). The covering map of the thrice punctured sphere can be described in terms of elliptic functions.

Theorem 4.20. The function

λ(τ) = \frac{e_{3} − e_{2}}{e_{1} − e_{2}} = \frac{℘ (\frac{1+τ}{2} ) − ℘ (\frac{τ}{2})}{℘ (\frac{1}{2}) − ℘ (\frac{τ}{2})}, τ ∈ ℍ^{2},

is a holomorphic universal covering map of the plane punctured at 0 and 1 whose group of deck transformations is Γ(2).

Proof. We only outline the proof since we will subsequently present different approaches to the issues that arise in the proof. The motion z ↦ \frac{z−e_{2}}{e_{1}−e_{2}} is a holomorphic map of the complex sphere taking e_{2} to the origin, e_{1} to the point 1, that preserves the point ∞. λ(τ) is the image of the point e_{3} under this map, which may be viewed as a change of parameters defining conformally equivalent tori. We use some of the notation introduced in the previous subsection. The function λ is holomorphic and Γ(2)-invariant; that is,

λ ∘ γ = λ, for all γ ∈ Γ(2).

To verify the claim about invariance, we observe that Γ essentially permutes

the half periods as a consequence of (1.5): For γ = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ PSL(2, ℤ),

(cτ + d)^{2}e_{1}(τ) = (cτ + d)^{2}℘ (\frac{1}{2}, τ) = ℘ (ˆ\frac{1}{2}, τ̂) = ℘ (\frac{1}{2}(a − cτ̂), τ̂) ,

which equals

\{ \begin{matrix}℘ (\frac{τ̂}{2}, τ̂) = e_{2}(τ̂) if a is even and c is odd \\ ℘ (\frac{1}{2}, τ̂) = e_{1}(τ̂) if a is odd and c is even \\ ℘ (\frac{1+τ̂}{2}, τ̂) = e_{3}(τ̂) if a is odd and c is odd \end{matrix} .

Note that it is impossible for both a and c to be even. Similarly,

(cτ + d)^{2}e_{2}(τ) = (cτ + d)^{2}℘ (\frac{τ}{2} , τ) = ℘ (\frac{τ̂}{2}, τ̂) = ℘ (\frac{1}{2}(−b + dτ̂), τ̂) ,

which equals

\{ \begin{matrix}℘ (\frac{τ̂}{2}, τ̂) = e_{2}(τ̂) if b is even and d is odd \\ ℘ (\frac{1}{2}, τ̂) = e_{1}(τ̂) if b is odd and d is even \\ ℘ (\frac{1+τ̂}{2}, τ̂) = e_{3}(τ̂) if b is odd and d is odd \end{matrix} ,

and

\begin{matrix}(cτ + d)^{2}e_{3}(τ) = & (cτ + d)^{2}℘ (\frac{1+τ}{2} , τ) = ℘ (\frac{ˆ1+τ̂}{2}, τ̂) \\ = & ℘ (\frac{1}{2}(a − b − (c − d)τ̂), τ̂) \end{matrix} ,

which equals

\{ \begin{matrix}℘ (\frac{1+τ̂}{2}, τ̂) = e_{3}(τ̂) if a − b is odd and c − d is odd \\ ℘ (\frac{1}{2}, τ̂) = e_{1}(τ̂) if a − b is odd and c − d is even \\ ℘ (\frac{τ̂}{2}, τ̂) = e_{2}(τ̂) if a − b is even and c − d is odd \end{matrix} .

An element γ ∈ Γ(2) (thus a and d are odd, while b and c are even) fixes each half period. It is not too hard to construct a fundamental domain for the action of Γ(2) on ℍ^{2} (it is described by the last figure; see also §6.4) and show that λ maps this fundamental domain injectively into ℂ − {0, 1}. We can avoid this argument by observing (we already know that λ is holomorphic and misses 0 and 1) that Γ(2) has three equivalence classes of parabolic fixed points, which can be taken to be ∞, 0, 1. Further, the function λ extends by continuity to the fixed points of the parabolic elements of the group Γ(2), that is, to the rational points and the ideal point at infinity. One can now show that

λ(∞) = 0, λ(0) = 1, λ(1) = ∞;

as a matter of fact

λ(τ) exp{−πıτ} → 16 as ℑτ → ∞.

(We will produce at least two alternate formulae for the function λ: one involving Poincaré series (we will only outline the theory needed for this approach), and the second derived from the Riemann θ-function (based on the main ideas discussed in this book).) From the last formula it will be easy to see that in terms of the local coordinate x = exp(πıτ),

λ(τ) = 16x(1 − 48x + O(x^{2})).

Finally (ℍ^{2} ∪ ℚ ∪ {∞})/Γ(2) is compact. Since λ is a nonconstant holomorphic map from this compact surface onto ℂ̂ with a single simple pole, it is surjective and a holomorphic universal covering of its image which must be ℂ − {0, 1}.

To obtain the map from the punctured sphere to the moduli space R(1, 0), we must find out how the permutation group acts on the thrice punctured sphere. One can show that Perm(3) is faithfully represented by the six Möbius transformations that permute the points 0, 1 and ∞. This is, in fact, the full automorphism group of the thrice punctured sphere ℂ̂ − {0, 1, ∞}. The maps

τ ↦ −\frac{1}{τ} and τ ↦ 1 − \frac{1}{τ}

generate PSL(2, ℤ) (thus the corresponding Γ(2)-cosets generate the quotient PSL(2, ℤ)/Γ(2)). The first of these maps interchanges zero and infinity and sends one to minus one (which is equivalent modulo Γ(2) to plus one). Hence its image in Perm(3) must be the map

λ ↦ 1 − λ;

that is,

λ (−\frac{1}{τ} ) = 1 − λ(τ).

Similarly, the image of the second map must be

λ ↦ 1 − \frac{1}{λ}.

Since fixed points of maps in the τ-plane project to fixed points of maps in the λ-plane, we have

λ(ı) = \frac{1}{2} and λ (\frac{1}{2}(1 + \sqrt{3}ı)) = \frac{1}{2}(1 ± \sqrt{3}ı).

(We do not determine which choice of sign is correct; it can be done easily.) In the above calculations we have used the fact that PSL(2, ℤ) is the normalizer of Γ(2), that Perm(3) is isomorphic to Aut(ℍ^{2}/Γ(2)), and that we have for each γ ∈ PSL(2, ℤ) the commutative diagram

ℍ^{2}  ⟶^{γ} ℍ^{2}
↓  ↓
ℍ^{2}/Γ(2) ⟶^{θ(γ)} ℍ^{2}/Γ(2)

To get a function invariant under the full modular group, we take any rational function of λ with a single simple pole at a point that is not fixed by any element in the group Perm(3) (that is, a rational function of degree one with a pole at a point not fixed by the group of Möbius transformations that permute the three points 0, 1, ∞) (for example, we can use the function λ ↦ \frac{1}{λ−3}) and average this function over the group Perm(3).^{19} This averaging process leads to the rational function

(\frac{1}{λ − 3} + \frac{λ}{1 − 3λ} − \frac{1}{λ + 2} + \frac{1 − λ}{3λ − 2} + \frac{λ − 1}{3 − 2λ} − \frac{λ}{2λ + 1}) .

To normalize, we postcompose the resulting average with a Möbius transformation C, so that the composite map sends λ = ∞ to the point ∞, λ = \frac{1+\sqrt{3}ı}{2} to 0 and λ = 2 to 1. The Möbius transformation C is computed to be (\frac{7}{9})^{2} \frac{7z+15}{3z+5}, and we obtain the famous ȷ-invariant

ȷ(λ) = \frac{4}{27} \frac{(1 − λ + λ^{2})^{3}}{λ^{2}(1 − λ)^{2}} .

The above may appear a bit mysterious and magical since we began with an arbitrary rational function with a simple pole at a nonfixed point of Perm(3) (at 3 in our case), averaged it over this finite group of order 6 to obtain a rational function of degree 6 with double poles at the orbit of 3 under the group. We then find the Möbius transformation taking the poles to 0, 1, ∞. It is perhaps the last part which is most mysterious, so we present an additional procedure which may be a little less mysterious.

In order to obtain a Γ-invariant function, we average a Γ(2)-invariant function over the group Γ(2)\Γ. In place of the function we used above we can average the function λ itself. This produces a constant function. So we try to use the next simplest case and average the function λ^{2}. This leads us to the rational function

F (λ) = \frac{2λ^{6} − 6λ^{5} + 9λ^{4} − 8λ^{3} + 9λ^{2} − 6λ + 2}{λ^{2}(1 − λ)^{2}}.

The function F has the virtue of having the correct poles. The reader can now check that

\frac{F (λ) + 3}{2} = \frac{(1 − λ + λ^{2})^{3}}{λ^{2}(1 − λ)^{2}} .

This still does not explain the constant \frac{4}{27}. We leave it to the reader to consult books on elliptic functions to discover the reasons for this normalization. Based on our work on θ-constants we will conclude that in terms of

(−) = λ, (0, 1) = 1 − λ, (0, ∞) = \frac{1}{λ} , (0, 1, ∞) = \frac{1}{1 − λ} , (0, ∞, 1) = \frac{λ − 1}{λ} , (1, ∞) = \frac{λ}{λ − 1} .

19Realized, as before, as the group of fractional linear transformations that permute the three points ∞, 0 and 1. If we identify the permutation σ ∈ Perm(3) with the motion in PSL(2,ℂ)

that sends λ to σ(λ) for λ = 0, 1 and ∞, then we summarize this identification by the entries σ = σ(λ) in We are using here standard permutation notation; see, for example, [9, Ch. 5].

the local coordinate x = exp(2πıτ),

ȷ(τ) = \frac{1}{4 · 27x}(1 − 4^{3}7x + O(x^{2})).

More important than the formulae produced is an understanding of the maps involved. We are studying the coverings

ℍ^{2} ⟶^{λ} ℂ − {0, 1} ≅ ℍ^{2}/Γ(2) ⟶^{ȷ} ℂ ≅ ℍ^{2}/Γ.

The group Γ(2) is torsion free (isomorphic to the fundamental group of the thrice punctured sphere ℂ − {0, 1}), and λ is a holomorphic universal covering map. The map ȷ ∘ λ is the branched universal covering map of the orbifold ℍ^{2}/Γ. Since Γ has only 2 and 3 torsion, the map ȷ must be branched. Viewing ȷ as a selfmap of the sphere, we see that ȷ has branch number 2 at ∞ as well as −1, \frac{1}{2} and 2; it has branch number 3 at e^{\frac{πı}{3}} and e^{\frac{5πı}{3}} . We conclude that ȷ sends the three punctures (\{0, 1, ∞\}) on ℍ^{2}/Γ(2) to ∞. The punctures on ℍ^{2}/Γ(2) are the image of ℚ ∪ {∞} under λ; the Γ-orbit of ı is sent by λ to \{−1, \frac{1}{2}, 2\} and by ȷ to 1; λ sends the Γ-orbit of \frac{1}{2} + \frac{\sqrt{3}}{2}ı = e^{\frac{πı}{3}} to \{\frac{1}{2} ± \frac{\sqrt{3}}{2}\} and by ȷ to 0.

Let us summarize what we have accomplished. For a point τ ∈ ℍ^{2}, we constructed a torus T_{τ} and its ȷ-invariant ȷ(λ(τ)) which is a complete conformal invariant for the family of tori in the following sense. Given two points τ_{1} and τ_{2} in ℍ^{2}, then T_{τ_{1}} is conformally equivalent to T_{τ_{2}} if and only if ȷ(λ(τ_{1})) = ȷ(λ(τ_{2})) and for each point α ∈ ℂ, there is a τ ∈ ℍ^{2} such that α = ȷ(λ(τ)). For almost all α ∈ ℂ (we shall see shortly that we are only excluding α in the segment (−∞, 1] ⊂ ℝ), we can find a unique (also canonical) τ in the interior of ω(PSL(2, ℤ)) with j(λ(τ)) = α. The complex plane is, of course, not compact. We would like to have a natural compactification of the space of tori. We need a point τ such that j(λ(τ)) = ∞. Such a point τ must hence be sent by λ to 0, 1 or ∞. Hence τ cannot belong to ℍ^{2}; it necessarily will belong to ℚ ∪ {∞}. We will see in a little while (§4.6) how to add the point at infinity to our space of moduli of surfaces of genus one.

Exercise 4.21. Show that for each γ ∈ Γ, there is a permutation σ_{γ} of the integers 1, 2 and 3 such that

(e_{i} ∘ γ)γ′ = e_{σ_{γ}(i)}, i = 1, 2, 3.

Further, the map γ ↦ σ_{γ} is a surjective homomorphism from Γ to the permutation group on three letters with kernel Γ(2).

4.4. Tori with symmetries. The group of conformal automorphisms of the torus T_{τ}, Aut T_{τ} , contains the subgroup of translations by elements of T_{τ}. Hence we should consider instead the factor group (Aut T_{τ})/T_{τ} which can be canonically identified with the (extra) symmetries of T_{τ}

Aut_{o}T_{τ} = {C ∈ Aut T_{τ}; C(0) = 0}.

It is best to encode the above information in a short exact sequence that splits

0 → Aut_{o}T_{τ} → Aut T_{τ} → T_{τ} → 0;

the second arrow is the inclusion map and the third arrow sends the automorphism C to the point C(0). The splitting map sends the point on T_{τ} determined by a ∈ ℂ to the translation of the torus induced by the affine map z ↦ z + a. We observe that every torus has at least one symmetry. It is given by the conformal involution Ẽ(z) = −z + 1 + τ with fixed point the half period \frac{1+τ}{2} . We are choosing this particular involution because it preserves the fundamental parallelogram ω̃ for G̃_{τ} . It is equivalent modulo T_{τ} to the motion z ↦ −z ∈ Aut_{o}T_{τ} . The involution has the property that it conjugates Ã and B̃ to their respective inverses. The generic torus has no other symmetries. The “square” torus (the one corresponding to τ = ı) has the additional symmetry corresponding to the motion of order four, z ↦ ız. The “rhombic” torus (τ = \frac{1+ı\sqrt{3}}{2}) has the extra symmetry of order six, z ↦ τz. We have accounted for two of the three distinguished points in the compactification of the moduli space R(1, 0). The third point is distinguished because it is missing from the moduli space R(1, 0). It is also distinguished for other reasons as we shall see shortly.

It is useful to describe tori with additional symmetries. The tori with purely imaginary moduli (ℜ(τ) = 0) have the anti-conformal involution J̃ given by

J̃(z) = z̄ + τ.

This involution fixes the point \frac{1+τ}{2} and conjugates Ã to itself and B̃ to its inverse.

Tori with moduli of absolute value 1 (|τ| = 1) have a different anticonformal involution J̃_{1} defined by

J̃_{1}(z) = τz̄.

This involution also fixes \frac{1+τ}{2} and conjugates Ã to B̃ and B̃ to Ã.

We will distinguish two more families of tori: those with ℜτ = −\frac{1}{2} and those with ℜτ = \frac{1}{2}. Each member of the first (second) family has the involution J̃_{3} (J̃_{2}) z ↦ −z̄ (z ↦ −z̄ + 1) that conjugates Ã to its inverse and B̃ to Ã ∘ B̃ (Ã to its inverse and B̃ to Ã^{−1} ∘ B̃).

The families of tori discussed above lie on lines in ℍ^{2} determined by segments on the boundary of ω(PSL(2, ℤ)) and on the axis of symmetry of the fundamental domain we have chosen for PSL(2, ℤ)).^{20} The one symmetry that is present on all tori can be described more intrinsically. Given a torus T and a point x on it, there exists a unique conformal involution Ẽ of T that fixes x. If we now identify x with the origin of T, then Ẽ fixes, in addition to 0, the other three half periods (points of order 2) of T.

We can use symmetries to obtain information about the ȷ-invariant. The Weierstrass ℘-function is the unique doubly periodic function of degree two with Laurent series expansion at the origin of the form

℘(z) = \frac{1}{z^{2}} + ∑_{n=1}^{∞} a_{n}z^{2n}.

The function ℘ is an even function of z. However, the coefficients appearing in the Laurent series expansion are not in general real. If for τ in the upper half plane we have τ = −τ̄, then the torus T_{τ} is invariant under the involution z ↦ z̄, and we conclude from the uniqueness of the Laurent series expansion (or the original formula) for ℘ that \overline{℘(z̄)} = ℘(z) for all z ∈ ℂ. It follows that e_{1} ∈ ℝ. Also

e_{2} = ℘ (\frac{τ}{2} ) = ℘ (\frac{τ̄}{2}) = \overline{℘ (\frac{τ}{2} )} = \overline{e_{2}}.

Similarly we see that e_{3} is also real. Hence (if τ is imaginary) λ(τ) ∈ ℝ and ȷ(τ) ∈ ℝ. Thus we see that

τ = −τ̄ ⟹ λ = λ̄ and ȷ = j̄.

The segment {ıy ∈ ℂ; y > 1} gets mapped injectively by j ∘λ to the interval [1, ∞) ⊂ ℝ. We consider τ ∈ ℍ^{2} with |τ| = 1. The torus T_{τ} is invariant under the map z ↦ τz̄. The Laurent series for the ℘-function now shows that

℘(τz̄) = τ^{−2}\overline{℘(z)}, z ∈ ℂ.

We conclude that

℘ (\frac{τ}{2} ) = τ^{−2}\overline{℘ (\frac{1}{2})},

and

τ^{−2}\overline{℘ (\frac{1}{2}(1 + τ))} = ℘ (\frac{1}{2}(1 + τ)) .

One easily computes from these identities that

|τ| = 1 ⟹ λ + λ̄ = 1.

We conclude once again that λ = \frac{1}{2} for τ = ı and that ȷ is real for τ of absolute value one. The function j∘λ maps the circular segment \{exp ıθ; \frac{π}{3} ≤ θ ≤ \frac{π}{2} \} ⊂ ℂ injectively to [−\frac{4}{27}, 1] ⊂ ℝ. It is also easy to see that λ takes

20The fundamental domain ω(PSL(2, ℤ)) is invariant under the motion z ↦ −z̄.