Modular Forms

The following sections are included:

• Acknowledgements

• Contents

The following sections are included:

• This book

• Possible courses

• An overview of this book

The following sections are included:

• 18^th Century — a prologue

• 19^th century — the classical period

• Early 20^th century — arithmetic applications

• Later 20^th century — the link to elliptic curves

• The 21^st century — the Langlands Program

The following sections are included:

• Modular forms for SL₂(Z)

• Eisenstein series for the full modular group

• Computing Fourier expansions of Eisenstein series

• Congruence subgroups

• Fundamental domains

• Modular forms for congruence subgroups

• Eisenstein series for congruence subgroups

• Derivatives of modular forms
  • Quasi-modular forms

• Exercises

The following sections are included:

• Spaces of modular forms are finite-dimensional

• Explicit formulae for the dimensions of spaces of modular forms
  • Formulae for the full modular group
  • Formulae for congruence subgroups

• The Sturm bound

• Exercises

The following sections are included:

• Hecke operators
  • Motivation for the Hecke operators
  • Hecke operators for M_k(SL₂(Z))
  • Hecke operators for congruence subgroups

• Bases of eigenforms
  • The Petersson scalar product
  • The Hecke operators are Hermitian
  • Integral bases

• Oldforms and newforms
  • Multiplicity one for newforms

• Exercises

The following sections are included:

• Modular functions

• η-products and η-quotients

• The arithmetic of the j-invariant
  • The j-invariant and the Monster group
  • “Ramanujan's Constant”

• Applications of the modular function λ(z)
  • Computing digits of π using λ(z)
  • Proving Picard's Theorem

• Identities of series and products

• The Ramanujan-Petersson Conjecture

• Elliptic curves and modular forms
  • Fermat's Last Theorem

• Theta functions and their applications
  • Representations of n by a quadratic form in an even number of variables
  • Representations of n by a quadratic form in an odd number of variables
  • The Shimura correspondence

• CM modular forms

• Lacunary modular forms

• Exercises

The following sections are included:

• Classical treatment
  • The structure of the ring of mod p forms
  • The θ operator on mod p modular forms
  • Hecke operators and Hecke eigenforms

• Galois representations attached to mod p modular forms

• Katz modular forms

• The Sturm bound in characteristic p

• Computations with mod p modular forms

• Exercises

The following sections are included:

• Historical introduction to computations in number theory

• MAGMA
  • MAGMA philosophy
  • MAGMA programming

• SAGE
  • SAGE philosophy
  • SAGE programming
  • The SAGE interface
  • SAGE graphics

• PARI and other systems
  • PARI
  • Other systems and solutions

• Discussion of computation
  • Computation today
  • Expected running times
  • Using computation effectively
  • The limits of computation
    • Explicit examples of limitations
  • Guy's law of small numbers
  • How hard is it to calculate Fourier coefficients of modular forms?

• Exercises
  • MAGMA
  • SAGE
  • PARI
  • MAPLE

The following sections are included:

• Appendix A: Magma code for classical modular forms

• Appendix B: Sage code for classical modular forms

• Appendix C: Hints and answers to selected exercises

• Bibliography

• List of Symbols

• Index