Supersingular K3 surfaces for large primes
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1203.2889.pdf
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455.29 KB
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Author(s)
Maulik, Davesh
Date Issued
October 2014
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
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1215/00127094-2804783
