Contents

4.2. Characterization of G(k) 110

4.3. The surface M2/G(k) 112

4.4. Invariant classes for ro(fc) 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 Pieard's theorem 126

6.1. Reality conditions 127

6.2. Hyperbolicity and Picard's theorem 128

§7. Spaces of iV-th order ^-functions 129

§8. The Jacobi triple product identity 138

8.1. The triple product identity 138

8.2. The quintuple product identity 143

Chapter 3. Function theory for the modular group T and its 147

subgroups

§1. Automorphic forms and functions 148

1.1. Two Banach spaces 148

1.2. Poincare series 151

1.3. Relative Poincare series 152

1.4. Projections to the surface 154

1.5. Factors of automorphy 157

1.6. Multiplicative q-forms 159

1.7. Residues 161

1.8. Weierstrass points for subspaces of

A(H2,

(7, e) 162

§2. Automorphic forms constructed from theta constants 165

2.1. The order of automorphic forms at cusps - Fourier series

expansions at too 165

2.2. Automorphic forms for T(k) 172

2.3. Meromorphic automorphic functions for T(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 T0(k) 178

2.7. The structure of 0^

o

A

g

(H

2

5

T) and e^

0

A+(H

2

?

T) 179