Comparison isomorphisms for smooth formal schemes

Abstract

Contents 1.2 NotationsLet p > 0 be a prime integer, O K a complete discrete valuation ring with fraction field K and perfect residue field k. Fix an algebraic closure K ⊂ K, let k denote its residue field and O K the normalization of O K in K. Write G K for the Galois group of K over K. Fix a field extension K ⊂ M ⊂ K. We write M 0 ⊆ M for the maximal absolutely unramified subfield of M and O M 0 for its ring of integers.The following notations will be used throughout the paper (some of the objects denoted here will be defined in this very section and the rest in the next sections):

The Tate conjecture for K3 surfaces over finite fields

“…By results of Beauville-Donagi in [5], given an embedding of K into C, the variety F C is of K3 [2] type. If q is the Beauville-Bogomolov form, and h is the ample class on F C is the ample class corresponding to the Plücker embedding, then q(h) = 6.…”

“…Let S be a supersingular K3 surface over a finite field k of characteristic at least 5. Let S [2] be the Hilbert scheme that parametrizes length 2 zero-dimensional subschemes of S. By [18], S [2] is a smooth projective variety of dimension 4.…”

“…The variety S [2] is the quotient of the blow-up of S × S along the diagonal by the involution exchanging the two factors. By [4,Proposition 6], the second cohomology group of S [2] is generated by the second cohomology group of S and the class [E] of the exceptional divisor.…”

“…By [4,Proposition 6], the second cohomology group of S [2] is generated by the second cohomology group of S and the class [E] of the exceptional divisor. As a consequence, S satisfies the Tate conjecture if and only if S [2] satisfies the Tate conjecture for divisors.…”

“…By [14], S lifts to a projective K3 surface S over the ring W of Witt vectors of k. The variety S [2] lifts to the relative Hilbert scheme S [2] . Let K be the fraction field of W , and fix an embedding of K into C. Let q be the Beauville-Bogomolov form on H 2 (S [2] C , Z), and let h be an ample cohomology class.…”