Supersingular K3 surfaces for large primes

By Davesh Maulik. 2012

136 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1ea7d701a2562f7b0439c483cb7abf81a050ef37

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Citations in the corpus

Lectures on K3 Surfaces (2016, methodology): “This has been achieved by Maulik and Madapusi Pera in [385, 396], see also Benoist’s Bourbaki survey [57].”
The Tate conjecture for K3 surfaces over finite fields (2012, methodology): “As a consequence, we do not use any of the birational arguments of [23].”
Curves and cycles on K3 surfaces (2013, background): “d a curve C ⊂ X over a ﬁnite ﬁeld Fq. Suppose Kimura–O’Sullivan ﬁnite-dimensionality holds for X and Ce and hence for X ×Ce (cf. [33, Prop. 5.10]). Then the Tate conjecture T1 for X (using the recent [13, 37, 39]) and Ce, which implies the Tate conjecture T1 for X × Ce and by duality the Tate conjec”
Supersingular K3 surfaces are unirational (2013, background): “upersingular, that is, their formal Brauer groups are of inﬁnite height. It follows from recent progress in the Tate-conjecture for K3 surfaces due to Charles [Ch12], Madapusi Pera [MP13], and Maulik [Ma12] that a K3 surface in odd characteristic is Artin-supersingular if and only ifit is Shioda-sup”
2-ADIC INTEGRAL CANONICAL MODELS (2016, result): “Of course, the case when the field has characteristic 6= 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).”