PREFACE

The spirit of a three-year-old will be

kept until 100 — A Japanese saying

There are two principal themes in the present book. The first one concerns a

quadratic Diophantine equation of the form X^\-=1 (fijxixj — 7 ? where ip = (Pij)

is a symmetric matrix with coefficients in Z and 0 / g G Z. For given ip and q

a solution x = (x{) with Xi G Z is called primitive if the Xi have no nontrivial

common divisor. When the equation is of the form x\ +x\ + x\ = q, Gauss showed

in his Disquisitiones that the number of primitive solutions is an elementary factor

times the class number of primitive binary quadratic forms of discriminant —q. His

proof is roundabout; besides, though later researchers treated the cases of five and

seven squares and obtained the formula for the number of primitive representations

in terms of special values of Dirichlet L-functions, the connection with the class

number was never explained satisfactorily. In the present book I show that the

number of orbits of primitive solutions under a group of units is essentially the

class number of an orthogonal group of degree n — 1, in such a way that the fact

specialized to the case of three squares gives the result of Gauss in a more direct

way.

The other principal theme is a certain Euler product, the idea of which originated

in January 1961 when I presented a theory of Hecke operators for Sp(n, Q) in a

seminar at the University of Tokyo. Taking the group of similitudes instead of

Sp{n, Q), I found, when n — 2, an Euler product of degree 4, and also a congruence

relation for each Euler p- factor. The results were published in a short article two

years later, and I turned my attention to other directions since then. Though I was

always conscious of this problem, I was unable to gain a clear perspective until a

few years ago, when I realized that I could treat it as a special case of the theory for

Clifford groups. A large portion of this book is devoted to a full exposition of the

theory including meromorphic continuation of Euler products and Eisenstein series

on such groups, attaining my desire 42 years ago in a more general setting. At

the same time we treat Euler products and Eisenstein series on orthogonal groups,

which are actually special cases of those on Clifford groups. One of the new points

of this work is that the gamma factors can be determined explicitly when the

automorphic forms in question are nonholomorphic eigenfunctions of differential

operators.

vii