By Fernando Q. Gouvêa
There's now a wide physique of idea bearing on algebraic forms over finite fields, and plenty of conjectures during this sector are of serious curiosity to researchers in quantity conception and algebraic geometry. This booklet bargains with the mathematics of diagonal hypersurfaces over finite fields, with specific specialize in the Tate conjecture and the Lichtenbaum-Milne formulation for the relevant worth of the L-function. It combines theoretical and numerical paintings, and comprises tables of Picard numbers. even if this publication is geared toward specialists, the authors have incorporated a few historical past fabric to assist nonspecialists achieve entry to the consequences.
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Extra info for Arithmetic of Diagonal Hypersurfaces over Finite Fields
Then the following assertions hold: (Type I) Let a E 2Ln and let VA be the corresponding twisted Fermat motive of degree m and dimension n. Suppose that a E 91n+2 is an induced character of type I. Then the slopes of the Hodge polygon of the corresponding twisted Fermat motive VA increase by 1 from those of VA while keeping the same multiplicities. In other words, if VA has a Hodge slope j, 0 < j < n, with . multiplicity hi, then VA has a Hodge slope j + 1 with multiplicity h3. More generally, if a E %+2d is an induced character of type I, then the slopes of the Hodge polygon of the corresponding twisted Fermat motive increase by d from those of VA while keeping the same multiplicities.
Then VA inherits the structure of VA, that is, if VA is ordinary (resp. of Hodge-Witt type, resp. supersingular), then so is VA. All m-1 twisted Fermat motives VA of type I of dimension n + 2 branching out from the same VA of dimension n inherit the structure of VA. (Type II) Let a = (ao, ai, ... L" with an+2 + b2 = M. Suppose that a E +2 is an induced character of type II, say, a = a#b. Let VB denote the twisted Fermat motive corresponding to the (Z/mZ)x-orbit B of b. If both VA and VB are ordinary (resp.
2. There is a canonical exact sequence of connected smooth formal groups 0 -* UVn-, Dvn-, -> 0 where uvn-, is unipotent and Dvn-' is p-divisible, whose dimension and the height are explicitly given as follows: dim Dvn E ((i + 1) - AH(a)lf)) aE2('_ i