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Supersingular K3 surfaces for large primes

Author(s)
Maulik, Davesh
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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≥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≥5. 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.
Date issued
2014-10
URI
http://hdl.handle.net/1721.1/115923
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Duke Mathematical Journal
Publisher
Duke University Press
Citation
Maulik, Davesh. “Supersingular K3 Surfaces for Large Primes.” Duke Mathematical Journal, 163, 13 (October 2014): 2357–2425
Version: Original manuscript
ISSN
0012-7094
1547-7398

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