Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic

Abstract

Kuga and Satake associate with every polarized complex K3 surface (X, L) a complex abelian variety called the Kuga-Satake abelian variety of (X, L). We use this construction to define morphisms between moduli spaces of polarized K3 surface with certain level structures and moduli spaces of polarized abelian varieties with level structure over C. In this note we study these morphisms. We prove first that they are defined over finite extensions of Q. Then we show that they extend in positive characteristic. In this way we give an indirect construction of Kuga-Satake abelian varieties over an arbitrary base. We also give some applications of this construction to canonical lifts of ordinary K3 surfaces.

The Tate conjecture for K3 surfaces over finite fields

“…The following result is proved in [1,Theorem 8.4.3], see also [33] for the case of K3 surfaces. It follows from the fact that π : X → T admits a spin level n structure.…”

“…We can now use Proposition 6.1.2 of [33] to conclude that the Kuga-Satake mapping extends to T and get the following. The proof of Maulik can easily be adapted to our setting.…”

“…The Kuga-Satake mapping plays a major role in this paper. We refer to [23], as well as to the papers of André and Rizov [1,33], for most definitions and results. However, for reasons that will become clear below, we need to work with a slightly different definition as follows.…”

Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for K3 surfaces, and the Tate conjecture

“…Let n ≥ 3 be an integer prime to p. Up to replacing k by a finite extension whose degree only depends on n and the pair (Λ, l), we can assume that the family M → T is endowed with a spin structure of level n with respect to R 2 π * Z ℓ,prim . We refer to [Cha12, 3.2] and to [And96,Riz10,Mau12] for definitions and details.…”

Some Remarks on Non-Symplectic Automorphisms of K3 Surfaces Over a Field of Odd Characteristic

“…We say X is a nonsymplectic CM K3 surface of order N if rank T (X) is equal to φ(N ). A nonsymplectic CM K3 surface gives a CM point in a moduli Shimura variety, so it has a model over a number field ( [11]). It seems like that there are only few non-symplectic CM K3 surfaces.…”