Modular forms : basics and beyond

Includes index

Includes bibliography (p. 171-173) and references

ch. I Preliminaries 1.Symplectic groups and symmetric domains 2.Some algebraic and arithmetic preliminaries 3.Modular forms of integral weight ch. II Theta Functions and Factors of Automorphy 4.Classical theta functions 5.Modular forms of half-integral weight 6.Holomorphic and nonholomorphic modular forms on 1 ch. III The Rationality and Eisenstein Series 7.The rationality of modular forms 8.Dirichlet series and Eisenstein series 9.Eisenstein series as automorphic eigenforms ch. IV The Correspondence between Forms of Integral and Half-integral Weight 10.Theta series of indefinite quadratic forms 11.Theta integrals 12.Main theorems on the correspondence 13.Hecke operators ch. V The Arithmeticity of Critical Values of Dirichlet Series 14.The theory on SL2(Q) for integral weight 15.The eigenspaces of Rp 16.Main theorems on arithmeticity 17.Hilbert modular forms

This is an advanced book on modular forms. While there are many books published about modular forms, they are written at an elementary level, and not so interesting from the viewpoint of a reader who already knows thebasics. This book offers something new, which may satisfy the desire of such a reader. However, we state every definition and every essential fact concerning classical modular forms of one variable. One of the principal new features of this book is the theory of modular forms of half-integral weight, another being the discussion of theta functions and Eisenstein series of holomorphic and nonholomorphic types.