The name is absent

SOME OBSERVATIONS ON THE SIEGEL FORMULA

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 Φ₀ΘΦ_oo
with Φ₀,Φ_oo i∏ the Schwartz-Bruhat spaces y^,(λ'₀),y^,(X₀₀) for the obvious
decomposition X_a = X₀×I_a, we have

n- 1

Z(s, Φ) = elementary factor ∙ ∩ ζ(s- 2ι)∙

i = 0

f Φ_tβ(x) ∙ ∣P∕(x)jζ-<²"-¹>∙ ∣⅛,
vX_co

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.

REFERENCES

[1] Borel, A., Linear Algebraic Groups, Benjamin (1969).

[2] Chevalley, C., Séminaire sur la classification des groupes de Lie
algébriques, 2 vol., Paris (1958).

[3] Mars, J. G. M., Les nombres de Tamagawa de certains groupes ex-
ceptionnels, Bull. Soc. Math. France 94 (1966), 97-140.

[4] Ono, T., Gauss transforms and zeta-functions, Ann. Math. 91 (1970),
332-361.

[5] Serre, J.-P., Cohomologie Galoisienne, Springer-Verlag (1965).

[6‘] Siegel, C. L., Gesammelte Abhandlungen, I-III, Springer-Verlag (1966).

[7] Weil, A., Adeles and algebraic groups (Lect. Notes), Institute for
Advanced Study, Princeton (1961).

[<S] -------, Sur la formule de Siegel dans la théorie des groupes classiques,

Acta Math. 113 (1965), 1-87.

[P] Weyl, H., The Classical Groups_jTheir Invariants and Representations
Princeton (1946).

The Johns Hopkins University

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