5.4. Geometric interpretations 45

5.5. The period of a punctured torus 47

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. Poincare metric on simply connected domains 50

6.3. Fundamental domains 52

6.4. The principal congruence subgroups T(k) 54

6.5. Adjoining translations: The subgroups G(k) 62

6.6. The Hecke subgroups T0(k) 63

6.7. Structure of r(Jfe, 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 87

§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 T(k) 97

2.5. Punctures on

M2/T(k)

and the classes X0(k) 98

2.6. The classes in X0(k) 100

2.7. Invariant quadruples 105

2.8. Towers 105

§3. Punctures and characteristics 106

3.1. A correspondence 106

3.2. Branching 106

§4. More invariant classes 107

4.1. Invariant classes for G(k) 107