Modular forms : a classical approach / Henri Cohen, Fredrik Strömberg
- Author:
- Cohen, Henri
- Published:
- Providence, Rhode Island : American Mathematical Society, [2017]
- Physical Description:
- xii, 700 pages ; 27 cm.
- Additional Creators:
- Strömberg, Fredrik, 1973-
- Series:
- Contents:
- Machine generated contents note: 1.1.Modularity -- 1.2.Examples of Modular Forms from Different Sources -- 1.3.Notation -- Exercises -- 2.1.Elliptic Functions -- 2.2.Elliptic Curves -- 2.3.Theta Functions -- 2.4.Concluding Remarks -- Exercises -- 3.1.Classical Analytic Tools -- 3.2.Bessel Functions -- 3.3.Bernoulli Numbers and the Gamma and Zeta Functions -- 3.4.Classical Arithmetic Tools -- 3.5.The Lipschitz Summation Formulas -- 3.6.Whittaker functions -- Exercises -- 4.1.The Extended Upper Half-Plane -- 4.2.The Modular Group -- 4.3.Fundamental Domains of the Modular Group -- 4.4.Topology of Gamma -- 4.5.Metrics and Measures on Gamma -- 4.6.Fuchsian Groups of the First Kind -- Exercises -- 5.1.Introduction -- 5.2.Examples of Modular Forms: Eisenstein Series -- 5.3.Differential Operators -- 5.4.Taylor Coefficients of Modular Forms -- 5.5.Modular Forms on the Modular Group and Its Subgroups -- 5.6.Zeros, Poles, and Dimension Formulas -- 5.7.The Modular Invariant j -- 5.8.The Dedekind eta-Function and the Product Formula for Delta -- 5.9.Eta Quotients -- 5.10.A Brief Introduction to Complex Multiplication -- Exercises -- 6.1.Basic Tools -- 6.2.Subgroups of Gamma -- 6.3.Action on P1 (Q): Cusps -- 6.4.Action on -- 6.5.Sets of Integer Matrices of Given Determinant -- 6.6.The Atkin-Lehner Involutions WQ -- Exercises -- 7.1.General Definitions -- 7.2.The Case G = Gammao(N) -- 7.3.Links Between Forms on Different Groups -- 7.4.Dimensions of Spaces of Modular Forms on Gammao(N) -- 7.5.Computational Aspects -- Exercises -- 8.1.Definitions -- 8.2.Basic Results on Poincare and Eisenstein Series -- 8.3.Poincare and Eisenstein Series for Congruence Subgroups -- 8.4.Fourier Expansions -- 8.5.Eisenstein and Poincare Series in Mk(Gammao(N), X) -- 8.6.Generalization of the Petersson Scalar Product -- Exercises -- 9.1.Introduction -- 9.2.The Hecke Bounds for Fourier Coefficients -- 9.3.Kloosterman Sums and Applications -- 9.4.Petersson Products Involving Eisenstein Series -- 9.5.A Theorem of Siegel -- Exercises -- 10.1.Introduction -- 10.2.Introduction to Hecke Operators -- 10.3.The Hecke Operators Are Hermitian -- 10.4.Eigenvalues and Eigenfunctions of Hecke Operators on Gamma -- 10.5.Double Coset Operators -- 10.6.Bases of Modular Forms for the Full Modular Group -- 10.7.Euler Products -- 10.8.Convolutions -- Exercises -- 11.1.Introduction -- 11.2.The Main Theorem -- 11.3.Weil's Theorem -- 11.4.Application to the Riemann Zeta Function -- 11.5.Periods and Antiderivatives of Modular Forms -- 11.6.The Case of Eisenstein Series -- 11.7.Transformation under an Arbitrary gamma belongs to Gamma -- 11.8.Eichler Cohomology -- 11.9.Interpretation in Terms of Periods -- 11.10.Action of Hecke Operators on Periods -- 11.11.Rationality and Parity Theorems -- 11.12.Rankin-Selberg Theory -- Exercises -- 12.1.Nonholomorphic Eisenstein Series -- 12.2.The Spaces M(jk) and M(epsilonk) -- 12.3.Unfolding -- 12.4.Kernels and the Trace Formula -- 12.5.Generalization of Haberland's Formula -- 12.6.Computation of Petersson Inner Products -- Exercises -- 13.1.Introduction -- 13.2.Preliminary Results -- 13.3.The Theory of Newforms -- 13.4.Diagonalizing the Full Space of Modular Forms -- 13.5.The Trace Formula for Newforms -- 13.6.Computing Spaces of Modular Forms -- Exercises -- 14.1.Introduction and Motivation -- 14.2.The Fundamental Theorem -- 14.3.Lattices and Theta Functions -- 14.4.Vector-Valued Modular Forms and Representations of the Modular Group -- 14.5.Finite Quadratic Modules and Weil Representations -- Exercises -- 15.1.Modular Forms of Half-Integral Weight -- 15.2.Jacobi Forms -- 15.3.Maass Forms -- 15.4.Hilbert Modular Forms -- 15.5.Bianchi Modular Forms -- 15.6.Siegel Modular Forms -- Exercises.
- Subject(s):
- ISBN:
- 9780821849477 (hardcover : alkaline paper)
0821849476 (hardcover : alkaline paper) - Bibliography Note:
- Includes bibliographical references and indexes.
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