By Fernando Q. Gouvêa

There's now a wide physique of conception pertaining to algebraic types over finite fields, and lots of conjectures during this zone are of significant curiosity to researchers in quantity conception and algebraic geometry. This e-book offers with the mathematics of diagonal hypersurfaces over finite fields, with specific specialize in the Tate conjecture and the Lichtenbaum-Milne formulation for the critical worth of the L-function. It combines theoretical and numerical paintings, and comprises tables of Picard numbers. even though this e-book is geared toward specialists, the authors have integrated a few history fabric to aid nonspecialists achieve entry to the implications.

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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).