For the most part, this book is the translation from Japanese of the earlier book written jointly by Koji Doi and the author who revised it substantially for the English edition. It sets out to provide the reader with the basic knowledge of elliptic modular forms necessary to understand the recent developments in number theory. The first part gives the general theory of modular groups, modular forms and Hecke operators, with emphasis on the Hecke-Weil theory of the relation between modular forms and Dirichlet series. The second part is on the unit groups of quaternion algebras, which are seldom dealt with in books. The so-called Eichler-Selberg trace formula of Hecke operators follows next and the explicit computable formula is given. In the last chapter, written for the English edition, Eisenstein series with parameter are discussed following the recent work of Shimura: Eisenstein series are likely to play a very important role in the future progress of number theory, and this chapter provides a good introduction to the topic.

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i. Introductory remarks 1 Plato, but not Socrates, concluded that the Forms are substances. Whether the Forms are substances is not an issue that Socrates had in mind. He did not deny it, but neither did he affirm it. If Socrates were asked a series of questions designed to determine whether he believed that the Forms are substances, he would admit that he had no opinion about this philosophical issue. Unlike Plato, Socrates was not a metaphysician. The same, of course, would not have always been true of Plato. Unlike Socrates, he was a metaphysician. At some point in his career, and at least by the time of the Phaedo and the Republic, Plato did what Socrates never thought to do. Plato considered the question and concluded that the Forms are substances. Although this development occurred more than two thousand years ago, time has not eclipsed its importance. It is one of the most seminal events in the history of the philosophy. With his defense of Socrates's method of intellectual inquiry, and the development of his Theory of Forms, Plato caused a now familiar cluster of metaphysical and epistemological issues to become central to philosophy.