The author offers a complete collection of definitions, formulas, and proofs as required for modular forms in one variable, particularly for Hecke operators (where the trace formulas are painstakingly derived). This book serves as a valuable source and handy secondary reference for results. Indeed, the methods and terminology are of a (minimal) handbook style, without excessive abstraction. The chapters are on Fuchsian groups, Automorphic forms, $L$-functions, Modular groups and forms, Unit groups of quaternion algebras, Traces of Hecke operators, and Eisenstein series. This is a Japanese-to-English translation of part of a joint book with \n K. Doi\en \ref[ Automorphic forms and number theory (Japa-\break nese), Kinokuniya, Tokyo, 1976; per bibl.].

There are three accompanying tables. Table A lists the dimensions of cusp forms for $\Gamma_0(N)$ with even weights $2\leq k\leq 50$ and level $1\leq N\leq 50$ and prime $50\leq N\leq 100$. The same is done for newforms. A shorter supplementary table does the same for prime $N$ and with quadratic $\chi=(*/N)$. Table B has eigenvalues and characteristic equations of $T(p)$ for some primes $11\leq N\leq 71$ and weight $2$ (but $\chi=1$). Table C is a specialization of Table B to the cases $N=29,37$, with quadratic $\chi$. The tables, attributed to \n Y. Maeda\en, \n H. Wada\en, and \n N. Iwasaki\en, are covered by formulas in the text but are not further explained or examined for interesting entries through a post-mortem.

The book is tightly written with no respite offered the reader by treating simple cases first or offering historical intuition. It is hard for the reviewer to imagine any novice learning by first exposure. It is easier to imagine the elation experienced by learning the material (perhaps more theoretically) from sources and then recognizing it here in more detailed and concrete form. The author does not claim to give a universal survey (surely not in the bibliography). He limits himself to the approach associated largely with G. Shimura, whose classic text "Introduction to the arithmetic theory of automorphic functions, Iwanami Shoten, Tokyo, 1971" might help prepare the reader initially. A very helpful glossary of symbols is provided at the end.

Reviewed by Harvey Cohn