INTRODUCTION

I. As we said in the Preface, the first principal theme of the book concerns a

quadratic Diophantine equation. To be precise, we consider a vector space V of

dimension n over a field F, take an F-bilinear symmetric form p : V x V — » F,

and put ip[x] — /?(x, x) for x G V. Then our problem is to study the solutions h

of the equation

(1) p[h] =

q

for a given q G

Fx.

Since our results on this question are uninteresting if n — 2,

we assume n 2 in this introduction. Let O^ denote the orthogonal group of p

and let SO^ — {a G O^ | det(a) = l } . We let these groups act on the right, so

that xa is the image of x G V under a G O^. For a fixed q G F x the elements

h £ V such that ip[h] — q form a single orbit under SO^, an easy consequence of

Witt's theorem. As our basic field F we take an arbitrary algebraic number field

or its localization at a prime ideal. Let g denote the maximal order in F in both

cases. We fix a g-lattice L in V such that (p[x\ Eg for every x G L.

Now it is natural to consider the solutions h contained in a fixed L. Put r —

{7 G SO^ I L7 = L}. Our first point is that the set of such solutions form a finite

number of orbits under JT. Namely, there exists a finite set {hi}r™zl such that

m

(2) {heL\ p[h] = q } = [_j hzF.

i=i

This is true in both local and global cases, as proved in Theorems 10.3 and 11.1.

This fact, if nontrivial, is merely our starting point. We can naturally ask

whether there are more refined results; we can also ask what happens if we re-

strict h to "primitive solutions." Here an element h is called primitive if, roughly

speaking, the components of h have no nontrivial common divisors, whose precise

meaning must be defined. To answer these, we have to impose some conditions.

The first one is the notion of maximality: we call L maximal if L is maximal among

the lattices on which (p takes values in g. Another is the consideration of the

ideal (p(h, L). Indeed, if g = Z and L = Z n , then h is primitive if and only if

ip(h, L) — Z. Therefore, we fix a maximal lattice L, q G F x , and an ideal b; we

then consider the set of h G V such that ip[h] — q and /?(/i, L) — b; we do not

require that h G L. Now our first main result is that if F is local, then the set of all

such h for given q and b, if nonempty, is a single orbit under T (Theorem 10.5).

The setting in the global case is the same; we fix a maximal lattice L, q in a

number field F, and a fractional ideal b. For an arbitrary g-lattice A in V we put

1