By Yorozu S.

We supply a generalization of the outcome received via C. Currais-Bosch. Weconsider the -operator linked to a transverse Killing box v on acomplete foliated Riemannian manifold . less than a definite assumption,we end up that, for every belongs to the Lie algebra of the linearholonomy team . a different case of our consequence, the model of the foliationby issues, implies the implications given by way of B. Kostant (compact case) andC. Currfis-Bosch (non-compact case).

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The upper row is the Kummer sequence for 3 ( m ) , based on the fact that 3 ( X ) is divisible. The first vertical arrow is surjective since 3 ( m ) has finite index in E(M,). A simple diagram chase shows that the map is surjective and has kernel isomorphic to ker(e). The exact sequence 11 Proof. Let M run over all finite extensions of F, contained in K . 2. ~ u Im(XM)/Im(XM)div t has order bounded by lE(m),l, where m is the residue field of M . Now Iml is bounded by assumption. Hence it follows that Im(XK)/Im(nK) is a finite group.

Philibert, Une preuve de la conjecture de Mahler-Manin, Invent. Math. 124 (1996), 1-9. J. Coates, G. McConnell, Iwasawa theory of modular elliptic curves of analytic rank at most 1, J. London Math. Soc. 50 (1994), 243-264. J. Coates, R. Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. 124 (1996), 129-174. J. Coates, S. Howson, Euler characteristics and elliptic curves, Proc. Nat. Acad. Sci. USA 94 (1997), 11115-11117. J. Coates, S. Howson, Euler characteristics and elliptic curves 11, in preparation.

Thus in this case, F, C C L, has a quotient ~ a l ( F l F , ) E Z,. Therefore, and hence X = Gal(L,/F,) X(F, I F ) 1if p splits in F/$. Notice that, since @/Fis abelian, the action of T = y - 1 on ~al(k:/F,) is trivial. Thus, X/TX is infinite. Now if one considers the A-module Y = A/( fi (T)ai),where f i (T) is irreducible in A, then Y/TY is infinite if and only if fi(T) is an associate of T. Therefore, if F is an imaginary quadratic field in which p splits and if F, is the cyclotomic Bpextension of F, then TI f (T), where f (T) is a generator of the characteristic ideal of X .

### A-operator on complete foliated Riemannian manifolds by Yorozu S.

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