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
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
= £(Li[z, mnu)).
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.
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