Modular Forms : A Classical and Computational Introduction.

By: Kilford, LloydPublisher: Singapore : Imperial College Press, 2008Copyright date: ©2008Description: 1 online resource (236 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9781848162143Subject(s): Forms, Modular.;Algebraic spacesGenre/Form: Electronic books. Additional physical formats: Print version:: Modular Forms : A Classical and Computational IntroductionDDC classification: 512.73 LOC classification: QA243 -- .K54 2008ebOnline resources: Click to View
Contents:
Intro -- Contents -- Acknowledgements -- Introduction -- This book -- Possible courses -- An overview of this book1 -- 1. Historical overview -- 1.1 18th Century - a prologue -- 1.2 19th century - the classical period -- 1.3 Early 20th century - arithmetic applications -- 1.4 Later 20th century - the link to elliptic curves -- 1.5 The 21st century - the Langlands Program -- 2. Introduction to modular forms -- 2.1 Modular forms for SL2(Z) -- 2.2 Eisenstein series for the full modular group -- 2.3 Computing Fourier expansions of Eisenstein series -- 2.4 Congruence subgroups -- 2.5 Fundamental domains -- 2.6 Modular forms for congruence subgroups -- 2.7 Eisenstein series for congruence subgroups -- 2.8 Derivatives of modular forms -- 2.8.1 Quasi-modular forms -- 2.9 Exercises -- 3. Results on finite-dimensionality -- 3.1 Spaces of modular forms are finite-dimensional -- 3.2 Explicit formulae for the dimensions of spaces of modular forms -- 3.2.1 Formulae for the full modular group -- 3.2.2 Formulae for congruence subgroups -- 3.3 The Sturm bound -- 3.4 Exercises -- 4. The arithmetic of modular forms -- 4.1 Hecke operators -- 4.1.1 Motivation for the Hecke operators -- 4.1.2 Hecke operators for Mk(SL2(Z)) -- 4.1.3 Hecke operators for congruence subgroups -- 4.2 Bases of eigenforms -- 4.2.1 The Petersson scalar product -- 4.2.2 The Hecke operators are Hermitian -- 4.2.3 Integral bases -- 4.3 Oldforms and newforms -- 4.3.1 Multiplicity one for newforms -- 4.4 Exercises -- 5. Applications of modular forms -- 5.1 Modular functions -- 5.2 η-products and η-quotients -- 5.3 The arithmetic of the j-invariant -- 5.3.1 The j-invariant and the Monster group -- 5.3.2 "Ramanujan's Constant" -- 5.4 Applications of the modular function λ(z) -- 5.4.1 Computing digits of π using λ(z) -- 5.4.2 Proving Picard's Theorem -- 5.5 Identities of series and products.
5.6 The Ramanujan-Petersson Conjecture -- 5.7 Elliptic curves and modular forms -- 5.7.1 Fermat's Last Theorem -- 5.8 Theta functions and their applications -- 5.8.1 Representations of n by a quadratic form in an even number of variables -- 5.8.2 Representations of n by a quadratic form in an odd number of variables -- 5.8.3 The Shimura correspondence -- 5.9 CM modular forms -- 5.10 Lacunary modular forms -- 5.11 Exercises -- 6. Modular forms in characteristic p -- 6.1 Classical treatment -- 6.1.1 The structure of the ring of mod p forms -- 6.1.2 The θ operator on mod p modular forms -- 6.1.3 Hecke operators and Hecke eigenforms -- 6.2 Galois representations attached to mod p modular forms -- 6.3 Katz modular forms -- 6.4 The Sturm bound in characteristic p -- 6.5 Computations with mod p modular forms -- 6.6 Exercises -- 7. Computing with modular forms -- 7.1 Historical introduction to computations in number theory -- 7.2 Magma -- 7.2.1 Magma philosophy -- 7.2.2 Magma programming -- 7.3 Sage -- 7.3.1 Sage philosophy -- 7.3.2 Sage programming -- 7.3.3 The Sage interface -- 7.3.4 Sage graphics -- 7.4 Pari and other systems -- 7.4.1 Pari -- 7.4.2 Other systems and solutions -- 7.5 Discussion of computation -- 7.5.1 Computation today -- 7.5.2 Expected running times -- 7.5.3 Using computation effectively -- 7.5.4 The limits of computation -- 7.5.4.1 Explicit examples of limitations -- 7.5.5 Guy's law of small numbers -- 7.5.6 How hard is it to calculate Fourier coe cients of modular forms? -- 7.6 Exercises -- 7.6.1 Magma -- 7.6.2 Sage -- 7.6.3 Pari -- 7.6.4 Maple -- 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.
Summary: This book presents a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to such diverse subjects as the theory of quadratic forms, the proof of Fermat's last theorem and the approximation of pi. It provides a balanced overview of both the theoretical and computational sides of the subject, allowing a variety of courses to be taught from it.
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Intro -- Contents -- Acknowledgements -- Introduction -- This book -- Possible courses -- An overview of this book1 -- 1. Historical overview -- 1.1 18th Century - a prologue -- 1.2 19th century - the classical period -- 1.3 Early 20th century - arithmetic applications -- 1.4 Later 20th century - the link to elliptic curves -- 1.5 The 21st century - the Langlands Program -- 2. Introduction to modular forms -- 2.1 Modular forms for SL2(Z) -- 2.2 Eisenstein series for the full modular group -- 2.3 Computing Fourier expansions of Eisenstein series -- 2.4 Congruence subgroups -- 2.5 Fundamental domains -- 2.6 Modular forms for congruence subgroups -- 2.7 Eisenstein series for congruence subgroups -- 2.8 Derivatives of modular forms -- 2.8.1 Quasi-modular forms -- 2.9 Exercises -- 3. Results on finite-dimensionality -- 3.1 Spaces of modular forms are finite-dimensional -- 3.2 Explicit formulae for the dimensions of spaces of modular forms -- 3.2.1 Formulae for the full modular group -- 3.2.2 Formulae for congruence subgroups -- 3.3 The Sturm bound -- 3.4 Exercises -- 4. The arithmetic of modular forms -- 4.1 Hecke operators -- 4.1.1 Motivation for the Hecke operators -- 4.1.2 Hecke operators for Mk(SL2(Z)) -- 4.1.3 Hecke operators for congruence subgroups -- 4.2 Bases of eigenforms -- 4.2.1 The Petersson scalar product -- 4.2.2 The Hecke operators are Hermitian -- 4.2.3 Integral bases -- 4.3 Oldforms and newforms -- 4.3.1 Multiplicity one for newforms -- 4.4 Exercises -- 5. Applications of modular forms -- 5.1 Modular functions -- 5.2 η-products and η-quotients -- 5.3 The arithmetic of the j-invariant -- 5.3.1 The j-invariant and the Monster group -- 5.3.2 "Ramanujan's Constant" -- 5.4 Applications of the modular function λ(z) -- 5.4.1 Computing digits of π using λ(z) -- 5.4.2 Proving Picard's Theorem -- 5.5 Identities of series and products.

5.6 The Ramanujan-Petersson Conjecture -- 5.7 Elliptic curves and modular forms -- 5.7.1 Fermat's Last Theorem -- 5.8 Theta functions and their applications -- 5.8.1 Representations of n by a quadratic form in an even number of variables -- 5.8.2 Representations of n by a quadratic form in an odd number of variables -- 5.8.3 The Shimura correspondence -- 5.9 CM modular forms -- 5.10 Lacunary modular forms -- 5.11 Exercises -- 6. Modular forms in characteristic p -- 6.1 Classical treatment -- 6.1.1 The structure of the ring of mod p forms -- 6.1.2 The θ operator on mod p modular forms -- 6.1.3 Hecke operators and Hecke eigenforms -- 6.2 Galois representations attached to mod p modular forms -- 6.3 Katz modular forms -- 6.4 The Sturm bound in characteristic p -- 6.5 Computations with mod p modular forms -- 6.6 Exercises -- 7. Computing with modular forms -- 7.1 Historical introduction to computations in number theory -- 7.2 Magma -- 7.2.1 Magma philosophy -- 7.2.2 Magma programming -- 7.3 Sage -- 7.3.1 Sage philosophy -- 7.3.2 Sage programming -- 7.3.3 The Sage interface -- 7.3.4 Sage graphics -- 7.4 Pari and other systems -- 7.4.1 Pari -- 7.4.2 Other systems and solutions -- 7.5 Discussion of computation -- 7.5.1 Computation today -- 7.5.2 Expected running times -- 7.5.3 Using computation effectively -- 7.5.4 The limits of computation -- 7.5.4.1 Explicit examples of limitations -- 7.5.5 Guy's law of small numbers -- 7.5.6 How hard is it to calculate Fourier coe cients of modular forms? -- 7.6 Exercises -- 7.6.1 Magma -- 7.6.2 Sage -- 7.6.3 Pari -- 7.6.4 Maple -- 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.

This book presents a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to such diverse subjects as the theory of quadratic forms, the proof of Fermat's last theorem and the approximation of pi. It provides a balanced overview of both the theoretical and computational sides of the subject, allowing a variety of courses to be taught from it.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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