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Modular forms : a classical and computational introduction /
"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 qu...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
London ; Hackensack, NJ :
Imperial College Press,
©2008.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 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 [symbol]. 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. 2.9. 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. 3.3. The Sturm bound. 3.4. Exercises
- 4. The arithmetic of modular forms. 4.1. Hecke operators. 4.2. Bases of eigenforms. 4.3. Oldforms and newforms. 4.4. Exercises
- 5. Applications of modular forms. 5.1. Modular functions. 5.2. [symbol]-products and [symbol]-quotients. 5.3. The arithmetric of the [symbol]-invariant. 5.4. Applications of the modular function [symbol]. 5.5. Identities of series and products. 5.6. The Ramanujan-Petersson conjecture. 5.7. Elliptic curves and modular forms. 5.8. Theta functions and their applications. 5.9. CM modular forms. 5.10. Lacunary modular forms. 5.11. Exercises
- 6. Modular forms in characteristic [symbol]. 6.1. Classical treatment. 6.2. Galois representations attached to mod [symbol] modular forms. 6.3. Katz modular forms. 6.4. The Sturm bound in characteristic [symbol]. 6.5. Computations with mod [symbol] modular forms. 6.6. Exercises
- 7. Computing with modular forms. 7.1. Historical introduction to computations in number theory. 7.2. MAGMA. 7.3. SAGE. 7.4. PARI and other systems. 7.5. Discussion of computation. 7.6. Exercises.