1 October 2014 Supersingular K3 surfaces for large primes
Davesh Maulik
Duke Math. J. 163(13): 2357-2425 (1 October 2014). DOI: 10.1215/00127094-2804783


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 p5. 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 p5.

The argument uses Borcherds’s 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.


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Davesh Maulik. "Supersingular K3 surfaces for large primes." Duke Math. J. 163 (13) 2357 - 2425, 1 October 2014. https://doi.org/10.1215/00127094-2804783


Published: 1 October 2014
First available in Project Euclid: 1 October 2014

zbMATH: 1308.14043
MathSciNet: MR3265555
Digital Object Identifier: 10.1215/00127094-2804783

Primary: 14J28
Secondary: 14G15

Keywords: Borcherds products , K3 surfaces , Kuga–Satake varieties , semistable reduction , supersingular varieties , Tate conjecture

Rights: Copyright © 2014 Duke University Press

Vol.163 • No. 13 • 1 October 2014
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