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Introduction Chapter 1. The modular group and elliptic function theory 1. Mobius transformations 2. Riemann surfaces 3. Kleinian groups 3.1. Generalities 3.2. The situation of interest 4. The elliptic paradise 4.1. The family of tori 4.2. The algebraic curve associated to a torus 4.3. Invariants for tori 4.4. Tori with symmetries 4.5. Congruent numbers 4.6. The plumbing construction 4.7. Teichmfiller and moduli spaces for tori 4.8. Fiber spaces - the Teichmuller curve 5. Hyperbolic version of elliptic function theory 5.1. Fuchsian representation 5.2. Symmetries of once punctured tori 5.3. The modular group 5.4. Geometric interpretations 5.5. The period of a punctured torus 5.6. The function of degree two on the once punctured torus 5.7. The quasi-Fuchsian representation 6. Subgroups of the modular group 6.1. Basic properties 6.2. Poincare metric on simply connected domains 6.3. Fundamental domains 6.4. The principal congruence subgroups F(k) 6.5. Adjoining translations: The subgroups G(k) 6.6. The Hecke subgroups Fo(k) 6.7. Structure of F(k,k) 6.8. A two parameter family of groups 7. A geometric test for primality Chapter 2. Theta functions with characteristics 1. Theta functions and theta constants 1.1. Definitions and basic properties 1.2. The transformation formula 1.3. More transformation formulae 2. Characteristics 2.1. Classes of characteristics 2.2. Integral classes of characteristics 2.3. Rational classes of characteristics 2.4. Invariant classes for ¦£(k) 2.5. Punctures on H2/¦£(k) and the classes Xo(k) 2.6. The classes in Xo(k) 2.7. Invariant quadruples 2.8. Towers 3. Punctures and characteristics 3.1. A correspondence 3.2. Branching 4. More invariant classes 4.1. Invariant classes for G(k) ¡­¡­ Chapter 3. Function theory for the modular group ¦£ and its subgroups Chapter 4. Theta constant identities Chapter 5. Partition theory: Ramanujan congruences Chapter 6. Identities related to partition functions Chapter 7. Combinatorial and number theoretic applications Bibliography Bibliographical Notes Index