Dedication

pp v-vi

Contents

pp vii-xv

Organization

pp xvi-xvi

Acknowledgements

pp xvii-xviii

1 - Introduction

pp 1-6

Part I - Modular forms and their variants

pp 7-8

2 - Elliptic functions

pp 9-42

3 - Modular forms for SL(2; Z)

pp 43-70

4 - Variants of modular forms

pp 71-94

5 - Quantum fields on a torus

pp 95-120

6 - Congruence subgroups and modular curves

pp 121-136

7 - Modular forms for congruence subgroups

pp 137-154

8 - Modular derivatives and vector-valued modular forms

pp 155-167

9 - Modular graph functions and forms

pp 168-186

Part II - Extensions and applications

pp 187-188

10 - Hecke operators

pp 189-207

11 - Singular moduli and complex multiplication

pp 208-222

12 - String amplitudes

pp 223-258

13 - Toroidal compactication

pp 259-273

14 - S-duality of Type IIB superstrings

pp 274-293

15 - Dualities in N = 2 super Yang–Mills theories

pp 294-331

16 - Basic Galois theory

pp 332-344

Part III - Appendix

pp 345-346

Appendix A - Some arithmetic

pp 347-361

Appendix B - Riemann surfaces

pp 362-377

Appendix C - Line bundles on Riemann surfaces

pp 378-390

Appendix D - Riemann #-functions and meromorphic forms

pp 391-402

Appendix E - Solutions to exercises

pp 403-454

References

pp 455-476