Supersingular K3 surfaces for large primes

Abstract

Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height) then its Picard rank is 22. Along with work of Nygaard-Ogus, this conjecture implies the Tate conjecture for K3 surfaces over finite fields with p \geq 5. We prove Artin's conjecture under the additional assumption that X has a polarization of degree 2d with p > 2d+4. Assuming semistable reduction for surfaces in characteristic p, we can improve the main result to K3 surfaces which admit a polarization of degree prime-to-p when p \geq 5. The argument uses Borcherds' construction of automorphic forms on O(2,n) to construct ample divisors on the moduli space. We also establish finite-characteristic versions of the positivity of the Hodge bundle and the Kulikov-Pinkham-Persson classification of K3 degenerations. In the appendix by A. Snowden, a compatibility statement is proven between Clifford constructions and integral p-adic comparison functors.Comment: Some minor edits made; German error fixed; comments still welcom

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.…”

“…In odd characteristic, it has been proved in [Mau12,Cha12], and independently in [MP13b]. The first of this proofs relies on results of Borcherds on the Picard group of Shimura varieties, while the second one uses construction of canonical models of certain Shimura varieties.…”

2-Adic Integral Canonical Models

“…Of course, the case when the field has characteristic = 2, this is already known by the results of the second author in [17], and also by earlier work by Maulik [21] and Charles [7] (among many others). The new ingredient here is the characteristic 2 case.…”

Picard numbers of quintic surfaces

“…The Tate conjecture is open in general, but it has recently been proved for K3 surfaces outside characteristic 2 by work of Maulik [Mau12], Charles [Cha13] and Madapusi Pera [Mad13] (based on the now classical case of elliptic K3 surfaces from [ASD73]). Independently of the validity of the Tate conjecture, the embedding (2.1) gives an upper bound for ρ(S) as follows: it is known that NS(S p ) is always generated by divisor classes defined over a finite extension of the ground field; this implies that all eigenvalues of Frobenius on the algebraic part inside H 2 et (S p , Q ℓ (1)) are roots of unity.…”