By Jean Pierre Serre

Jean-Pierre Serre is Professor on the Collège de France. He has written a few books, together with "Algebraic teams and sophistication Fields", "Local Fields", "Complex Semisimple Lie Algebras", "Linear Representations of Finite Groups", accrued Papers (3 volumes), and "Trees" released by way of Springer-Verlag.

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2 Cohomology groups of V = V, ,(c) The geometry and topology of V = Vm,(c) are closely linked to those of the Fermat variety X = Vn (1), to which it is of course isomorphic over the algebraic closure k. In fact, the phrase "geometric and topological invariants" of V usually refers to quantities depending only on the base-change of V to k, which are therefore independent of the twisting vector c = (co, cl, . . , of the defining equation for V. Some examples are various cohomological constructions.

But AH(a) = AH(a) + 1, so that slopes of these formal groups are equal. Therefore, the assertion follows, as over k slopes determine completely the structure of p-divisible groups. 5. 12. 6. 12 Let (m, n) _ (7, n) with n > 1. Let p be a prime such that p - 1 (mod 7). Let a = (1,1,1, 4) E 212. Then DO,2 has slope 0 with multiplicity 2. Let a = (1,1,1,1, 4, 6) E 217 be an induced character of type I. Then Dv4 has A slope 1 with multiplicity 2, while D13 has slope 0 with multiplicity 2 and this A is isomorphic to D°y'A over k.

V is said to be ordinary if each twisted Fermat motive VA is ordinary. 2. V is said to be supersingular if each twisted Fermat motive VA is supersingular. 3. V is said to be strongly supersingular if each twisted Fermat motive VA is strongly suupersingular. V is said to be of Hodge-Witt type if each twisted Fermat motive VA is of Hodge-Witt type. 14 Most diagonal hypersurfaces are of mixed type. One easy case, however, was noted above: diagonal hypersurfaces of degree m are ordinary when p = 1 (mod m).