Modular Forms: A Classical And Computational IntroductionThis 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. |
Contents
1 Historical overview | 5 |
2 Introduction to modular forms | 11 |
3 Results on finitedimensionality | 41 |
4 The arithmetic of modular forms | 57 |
5 Applications of modular forms | 93 |
6 Modular forms in characteristic p | 143 |
7 Computing with modular forms | 163 |
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Common terms and phrases
a b c d algorithm arithmetic basis of eigenforms Chapter compute congruence subgroup consider constant term cusp form cuspidal define definition dimension formulae Dirichlet character double coset E4 and E6 eigenforms Eisenstein series element elliptic curves example Exercise Fermat’s Last Theorem find a basis finite finite-dimensional form f form of weight forms for SL2(Z Fourier coefficients Fourier expansion function of weight fundamental domain given Hecke algebra Hecke operators Hecke operators Tn holomorphic on H identity integer and let ISBN j-invariant Koblitz Let f Magma Math Mathematics matrix mod p modular modular function modular group modulo newform notation number theory oldforms operators Tp positive integer prime number proof Proposition prove Ramanujan Ramanujan-Petersson conjecture result Sage Section Shimura Sk SL2(Z space of modular Sturm bound theory of modular theta functions To(N transforms correctly weakly modular write zero