2

INTRODUCTION

r(A) = {~/eSO^\A~/ = A},

A[q, b] = { x G V | tp[x] = q, ip(x, A) = b }.

Assuming L[q, b] ^ 0, we fix an element h of L[q, b]; we then denote by W the

orthogonal complement of Fh in V with respect to p, and by if the special or-

thogonal group of the restriction of p to VK; we view H as the subgroup of SO^

consisting of the elements acting trivially on Fh. Defining the adelizations SO^

and ifA of SO^ and H as usual, we can let SO^ act on the set of all ^-lattices in

V, and therefore we can define the genus of A with respect to SO^ and also the

genus of A with respect to ifA- We call these the SO^-genus and the if-genus of A,

respectively. Taking SO^ and H instead of SO^ and ifA, we can similarly define

the SO^-cl&ss and the if-class of A. Now our main result (Theorem 11.6) states:

(A) Let {Li}iej be a minimal subset of the H-genus of L with the property that

the H-genus of L is contained in \_]ieI {LiQi | a G SO*}. For each k G L%\q, b] we

can find a G SO* such that k = ha, as mentioned above. Then Lta~x belongs

to the H-genus of L. Moreover, assigning the H-class of

LoT1

to k, we obtain a

bijection of \_jieI (Li[q, b}/r(Li)) onto the set of all H-classes in the H-genus of

L. Consequently, putting D = {£ G ifA | L£, — L}, we obtain

(3)

#(H\HA/D)

= £(Li[z, mnu)).

iei

It can happen that I consists of a single element, which is of course the case if

the *SO^-genus consists of a single class. In such a case we have

(4) #{H\HA/D) = #(L[g, b]/r(L)),

which means that the number of solutions h of y?[/i] = q under the condition

cp(h, L) = b, counted modulo the group T(L), is exactly the class number of the

group H with respect to D.

This type of class number of H is defined relative to how H is embedded in SO^',

and cannot be determined solely by if. This must be so, because the quantity of

(3) or (4) depends on ip, q, b, and the genus of L, whereas H is determined by (p

and q. Still, it is natural to ask how #(H\Hj±/D) is related to the intrinsic class

number of H, that is, #(H\HA/J) with J = {£ G HA | A/f = A/}, where M is

a g-lattice in W that is maximal with respect to the restriction of p to W. This

does not depend on the choice of M. Though the question is highly nontrivial in

general, we can give at least a clear-cut answer in the case n — 3 (Theorem 12.3)

and also several formulas in the higher-dimensional case.

Let us now state some noteworthy consequences of the above theorem.

(B) Suppose n 7 and the class number of F in the narrow sense is odd;

suppose also that ip represents 0 nontrivially at an archimedean prime of F, and

the same holds for the restriction of y to W. Then #{L[q, b]/F(L)} 1.

This is a special case of a more general result (Theorem 12.1) applicable to the

case n 4 under some mild conditions.