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.

Some minor edits made; German error fixed; comments still welcome

References in corpus (2)

• The Tate conjecture for K3 surfaces over finite fields
• The Tate conjecture for K3 surfaces in odd characteristic

Cited by in corpus (28)

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• Good Reduction of K3 Surfaces
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• Twistor spaces for supersingular K3 surfaces
• Tautological classes on moduli space of hyperkähler manifolds
• Combinatorial degenerations of surfaces and Calabi--Yau threefolds
• Supersingular K3 Surfaces are Unirational
• 2-adic integral canonical models and the Tate conjecture in characteristic 2
• Relations in the tautological ring of the moduli space of K3 surfaces
• Automorphisms of minimal entropy on supersingular K3 surfaces
• Degeneration of Kummer surfaces
• Integral Canonical Models for Automorphic Vector Bundles of Abelian Type
• On the unirationality of supersingular K3 surfaces
• Semi-positivity in positive characteristics
• Arithmetic occult period maps
• On Irreducible Symplectic Varieties of ${K3}^{[n]}$-type in Positive Characteristic
• Curves and cycles on K3 surfaces
• Supersingular O'Grady varieties of dimension six
• Isogenies between K3 Surfaces over ${\bar{\mathbb{F}}}_p$
• The representations of the automorphism groups and the Frobenius invariants of K3 surfaces
• Irreducibility of geometric Galois representations and the Tate conjecture for a family of elliptic surfaces
• On the Brauer groups of fibrations
• On the number of Enriques quotients for supersingular K3 surfaces
• A Kulikov-Type Classification Theorem for a One Parameter Family of K3-Surfaces Over a p-ADIC Field and a Good Reduction Criterion
• Picard groups on moduli of K3 surfaces with Mukai models
• The moduli space of marked supersingular Enriques surfaces