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
the basics. 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.
Thus the book is presented so that the reader can learn such theories
systematically. Ultimately, we concentrate on the following two themes:
(I) The correspondence between the forms of half-integral weight and
those of integral weight.
(II) The arithmeticity of various Dirichlet series associated with
modular forms of integral or half-integral weight.