4

INTRODUCTION

(3) as an equality between these two types of masses (Theorem 13.12). Though this

can be done for an arbitrary ip, for simplicity let us assume here that F is totally

real and (p is totally definite. Then m(#, D) — ^ • [Hnr(Lj) : l] , where {Lj}

is a complete set of representatives for the iJ-classes in the genus of L. Now we

have

(5) Yl

tr(L

)

: X

l ~

l

*

L

^ b] = m(ff, D).

iei

If we take #{h G Li | cp[h] = q) in place of #L^[g, b] and the representatives for

S'O^-classes in the S'O'^-genus of L in place of {L^}^/, then the left-hand side of

(5) becomes the weighted average for which Siegel gave a product formula in terms

of local representation densities. Now the right-hand side of (5) is the mass of an

(n — 1)-dimensional orthogonal group H1 for which he also gave a product formula.

Thus (5) establishes the equality between these two quantities defined in completely

different ways. Formula (5) is different from (3) in several ways. For one thing, (5)

concerns the mass of H relative to D, while (3) gives the class number of H relative

to D.

In our recent papers, we gave explicit formulas for m(if, D) and also for the

number of representations of an integer as sums of squares. Combining (5) with

those results, we can prove (Theorem 13.14):

(F) Let n, p, and L be as in (D); let K = Q(v^) , K = ( - l ) ^ " 1 ^ , with a

square free positive integer q; let L(s, x)

De

the L-function of the primitive Dirichlet

character \ corresponding to K. Then

#L[q,

2~lZ]

=

An(g)g("-2)/2(27r)--L(m,

X

),

#£[?, Z]=c„(g) •#£[?,

2"1Z].

Here m = (n — l)/2, 0 An(q) G Z, and 0 cn(q) G Q; the numbers An(q) and

cn(q) are determined explicitly by q (mod 8) and n. For example,

A7(q) = 2

9

-3

2

- 7 and c7(q) = 0 if q - 3 (£ 4Z,

A9(q) = 2 9 -3 2 -5-1 7 and c9(q) = 1/136 if q - 5 G 8Z.

II. Let us now turn to our second principal theme, which concerns Euler products

and Eisenstein series on O^ and a Clifford group. To be explicit, given (V, ip) as

above, let A(V) denote the Clifford algebra of (V, (p), and A+(V) its subalgebra

consisting of the even elements. Then we define the even Clifford group

G+(V)

by

G+(V) = G+(V, ^p) = {ae A+(V)X \ CTXVCL = V}.

We have two homomorphisms

r : G+{V) - SO* and v : G+(V) - F

x

,

defined by xr(a) = a~1xa for x G V and a G G + (F), and ^(xi-*-x

m

) =

y?[xi] • • '(p[xm] for a?i G V. It is well known that r gives an isomorphism of

G+(V)/F

X

onto SO*. If we put G

1

^ ) = {a G G+(V) | i/(a) = l } , then

Gl{V)

is

the so-called spin group of (V, /?). This is not our main object of study, though we

shall discuss its basic properties in the first several sections.