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Remark. ' Their difference is represented by the element of H1(0) corresponding to the above 1cocycle 8"' - e". Finally, call F " the sheaf on C x C' defined by just replacing "q-th power" by "p-th power" in the definition of F. For F", Proposition 3 will be replaced by Proposition 3'. The canonical homomorphism H1(F") H1(0) is bijective. The canonical homomorphism H2(E) --+ H2(F0) is surjective and its kernel is of dimension 4(pf-I - l)(g - 1) over F,: where q = pf. -+ This will be used only as a remark.

Moreover if, for a moment, S denotes a large finite set of valuations on k, then @dpi = lim S n S F$,(i*)+c(- vES k, ii*) 1 di*l , ; Case 2. We shall take as X (the underlying vector space of) a simple Jordan algebra defined over k of quaternionic hermitian matrices of degree m 2 2 and as f(x) its norm form. In this case the codimension of Sf in f-'(0) is 5. v ; this can be proved by replacing f(x) by the Pfaffian of an. alternating matrix of degree 2m, which is permissible for almost all v. The details are given in [3], pp.

7], Lemma 1. We also have 0,. Proof. Since v is "good," F$ = 1 on o, and XO, has measure 1. F,* is in L1(k,) by assumption, we get + O h S ), Since for almost all v ; cf. 5. this implies Therefore we get This CRITERIA FOR uniformly in i and v. We are ready to prove Theorem 1 : first of all the series is absolutely convergent for every 0 in Y(X,). This follows from (C2), Lemma 3, and from the fact (proved in [3], [5]) that the series for every @ in 9'(X,). defines a continuous L1-function F , on k, with Fg as its Fourier transform; and for every 0 in Y(X,).

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