Introduction to Modular FormsFrom the reviews: "This book gives a thorough introduction to several theories that are fundamental to research on modular forms. Most of the material, despite its importance, had previously been unavailable in textbook form. Complete and readable proofs are given... In conclusion, this book is a welcome addition to the literature for the growing number of students and mathematicians in other fields who want to understand the recent developments in the theory of modular forms." #Mathematical Reviews# "This book will certainly be indispensable to all those wishing to get an up-to-date initiation to the theory of modular forms." #Publicationes Mathematicae# |
Contents
I | 3 |
III | 5 |
IV | 12 |
V | 14 |
VI | 16 |
VIII | 21 |
IX | 24 |
XI | 29 |
XL | 123 |
XLI | 126 |
XLII | 138 |
XLIV | 142 |
XLV | 151 |
XLVII | 153 |
XLVIII | 154 |
XLIX | 156 |
XII | 32 |
XIII | 35 |
XIV | 44 |
XV | 57 |
XVII | 61 |
XVIII | 65 |
XIX | 68 |
XX | 69 |
XXI | 73 |
XXII | 76 |
XXIII | 81 |
XXIV | 84 |
XXV | 85 |
XXVI | 88 |
XXVII | 89 |
XXVIII | 93 |
XXIX | 96 |
XXX | 101 |
XXXII | 105 |
XXXIII | 108 |
XXXIV | 111 |
XXXV | 112 |
XXXVI | 114 |
XXXVII | 118 |
XXXIX | 122 |
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Common terms and phrases
a₁ abelian analytic assume Bernoulli numbers Cartan subgroup Chapter commutative concludes the proof congruence subgroups conjugate contained coset cusp forms define definition denote Dirichlet character Dirichlet series dx dy eigenfunction eigenspaces elements equal Euler product exists F₁(N factor finite number form of weight formula fundamental domain G₁ Galois Hecke algebra Hecke operators Hence holomorphic I₁(N infinity invariant isomorphic isomorphic to A4 isotropy group lattice Let f Let G Manin matrix meromorphic modular forms modular symbol module multiplication notation obtain operators T(n p-adic p-integral Petersson scalar product polynomial positive integer power series properties proves the theorem q-expansion quadratic rational number relation relatively prime representation right-hand side Serre shows SL₂(Z space of cusp subgroup of PGL2(F sublattice subspace Suppose Theorem 3.1 theory upper half plane variables vector space w₂ whence write Z/NZ z₁ Σ Σ