A Néron–Ogg–Shafarevich criterion for K3 surfaces

Abstract

The naive analogue of the Néron–Ogg–Shafarevich criterion is false for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified ℓ‐adic étale cohomology groups, but which do not admit good reduction over K. Assuming potential semi‐stable reduction, we show how to correct this by proving that a K3 surface has good reduction if and only if Hnormalét2false(XK¯,Qℓfalse) is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain ‘canonical reduction’ of X. We also prove the corresponding results for p‐adic étale cohomology.

CM liftings of surfaces over finite fields and their applications to the Tate conjecture

“…As in the proof of [19,Theorem 2.4], the formal completion of along the special fibre is a formal scheme (rather than a formal algebraic space), and there is an isomorphism ( ) ad of adic spaces over , where ( ) denotes the adic generic fibre of . We remark that the isomorphsim ( ) Here the right-hand side is the set of homomorphisms from D( ) ( ) [ Here, the second isomorphism is induced by the base change D( / ) ( ) ⊗ cris D( / ) ( cris ), and the third isomorphism is induced by the quasi-isogeny .…”

“…Let denote the orthogonal complement of − in Λ 3 , where − is considered as an element in the third . Hence, is equal to 8 ⊕2 ⊕ ⊕2 ⊕ + , and its signature is (19,2). The following result is well known.…”

“…which is a quadratic space over Z. It is unimodular and its signature is (19,3). We fix a positive integer > 0.…”

Ordinary K3 surfaces over a finite field

“…1.3] is non-singular. It follows from the construction of this model that P (X ′ , L ′ ) = P (X, L) ⊗ OK O K ′ (see the end of section 6 in [3]), and hence P (X, L) is a smooth projective model X over O K .…”

“…(M1) the pairing −, − on M is unimodular, even, and of signature (3,19); (M2) F x, F y = q 2 x, y for every x, y ∈ M . From Deligne's proof of the Weil conjectures for K3 surfaces [6] we also know (M3) the endomorphism F of M ⊗ C is semi-simple and all its eigenvalues have absolute value q.…”