SOME OBSERVATIONS ON THE SIEGEL FORMULA
75
As for the zeta-function Z(s,Φ) of the morphism Pf, it has all the prop-
erties that we have mentioned. Furthermore, if Φ is of the form Φ0ΘΦoo
with Φ0,Φoo i∏ the Schwartz-Bruhat spaces y,(λ'0),y,(X00) for the obvious
decomposition Xa = X0×Ia, we have
n- 1
Z(s, Φ) = elementary factor ∙ ∩ ζ(s- 2ι)∙
i = 0
f Φtβ(x) ∙ ∣P∕(x)jζ-<2"-1>∙ ∣⅛,
vXco
in which ζ(s) is the Dedekind zeta-function of к and ∣ dx∣to, for instance,
denotes the product of ∣dx∣t, for all archimedian υ,s.
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The Johns Hopkins University