By Fernando Q. Gouvêa
There's now a wide physique of idea bearing on algebraic types over finite fields, and plenty of conjectures during this region are of significant curiosity to researchers in quantity thought and algebraic geometry. This publication bargains with the mathematics of diagonal hypersurfaces over finite fields, with specified concentrate on the Tate conjecture and the Lichtenbaum-Milne formulation for the imperative worth of the L-function. It combines theoretical and numerical paintings, and contains tables of Picard numbers. even if this e-book is geared toward specialists, the authors have integrated a few heritage fabric to aid nonspecialists achieve entry to the consequences.
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Additional resources for Arithmetic of Diagonal Hypersurfaces over Finite Fields
Our computations are consistent with the following closed formulas for the stable d-th combinatorial Picard number pd(V) of a diagonal hypersurface of dimension 2d. 4 Assume that m is prime and that p - 1 (mod m), and let V be a diagonal hypersurface of dimension n = 2d and of degree m. Then the following assertions hold: 1. Ford=1, we have p1(V) = 1 + (m - 1)(3m - 6). 2. For d = 2, we have p2(V) = 1 + 5(m - 1)(3m2 - 15m + 20). 3. For d = 3, we have p3(V) = 1 + 5 7(m - 1)(3m3 - 27m2 + 86m - 95).
Of Hodge-Witt type, resp. supersingular), then so is VA. 8. 10 Let VA be a twisted Fermat motive of degree m and of even dimension n = 2d. Then the following statements are equivalent: Arithmetic of Diagonal Hypersurfaces over Finite Fields 35 1. VA is ordinary and supersingular. 2. IIaII = d for every a E A. 3. hd,d (VA) = Bn (VA) Proof: Clear. The next result gives some information about motives that are ordinary and are not supersingular. 11 Let VA be a twisted Fermat motive of degree m and of even dimension n = 2d.
The Artin-Mazur forH' (V, Gm) of V. 5 There is a connected smooth formal group 4)"'-' over k whose Cartier module is isomorphic to H'-'(V, WSti). In particular, the Artin-Mazur functor cn = Hn(V, Gm) is representable by a connected smooth formal group over k (Dvn of dimension p9 (V). Furthermore, I)yn-i has the following properties: 1. V"' is isomorphic over k to the corresponding formal group of the Fermat variety, F F. 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