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
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