INTRODUCTION

3

Of

(C) Suppose n = 3, F = Q, ip[x] = ^ L i xh and L = Z3; take b = Z. Let

q = c2m with an odd integer c and a squarefree integer m such that m ^ 7

(mod 8). Let k be the class number of the order in Q(y/^q) of conductor cZ.

Then #{L[q, Z]/T(L)} = fc/e, where e = 1 if m = 3 (mod 4) or q 2, and

e = 2 otherwise.

This is a reformulation of the result of Gauss mentioned in the Preface. We

can derive similar results for more general ternary forms over any number field

(Theorem 12.3).

(D) Suppose F = Q and ip[x] = Yl7=i xl with n = 5, 7, or 9; let L be a

maximal Z-lattice in V = Q^ and q an odd prime number. If n 5, suppose q

coincides with the discriminant of K =

Q(\/K), K

—

{—l)^n~l^2q.

Then L[q, Z] ^

0, and #{L[q, Z]/T(L)} equals the class number of SO(4) with respect to the

stabilizer of a maximal lattice in Q^_i, where I/J = diag[ln_4, qls].

This is included in Theorem 12.14 which is applicable also to an even q.

There is one more classical topic which can be understood in our framework, that

is, the theory of binary forms ax2 + bxy + cy2. Such a form is said to be primitive if

a, 6, c are integers with no nontrivial common divisors. We view this as a special

case of (1) by taking V = {h G M2(F) |

th

— h} with an algebraic number field F,

and putting ip[h] — det(h) for h G V. We associate the element h = , ,

/

V to the form

ax2

+ bxy +

cy2;

then — ip[2h] =

b2—4ac,

which is the discriminant of

the binary form. For A = M2(£|)nV, an element q G Fx, and a fractional ideal b we

consider the elements h eV such that ip[h] = q and ip(h, A) = b, which constitute

the set A[/, b]. Let A be the group of all transformations x i— det (a) _ 1 -faxa with

a G GI/2(9). Then we can prove (Theorem 12.9):

(E) The number c • #{A[g, b]/Z\} times the class number of F equals the class

number of the order in the field F(^/^q) of discriminant gb

- 2

, where c = 1/2 if

all archimedean and nonarchimedean primes of F are unramified in F{yf^q), and

c—\ otherwise.

In the classical case in which F = Q and g = b = Z, this implies that the class

number of primitive binary forms of discriminant — q equals the class number of the

order of discriminant q in Q(y/—q), a fact first proved by Dedekind. Our methods

are more conceptual and completely different.

We shall also investigate a higher-dimensional version of (1), that is, an equation

hcp'lh

= q for a square matrix q of size m n; the desired solutions h are (raxn)-

matrices. We shall prove analogues of (2) in both local and global cases (Theorems

13.2 and 13.3). If m = n — 1, we can prove a formula for the number of solutions

h of ip[h] = q modulo the group of units, under a condition of the same type as

ip(h, L) = b, in terms of the number of certain classes in the orthogonal group of

q (Theorem 13.10).

We can actually reformulate (3) by introducing the notion of the mass. With H

and D as above we can define the mass of H relative to D, written m(iJ, D); see

[S97, §24.1]. Now in §13.11 we shall define the mass of the set A[q, b] and express