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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2012
- DOI:
- 10.48550/arXiv.1203.2889
- arXiv:
- arXiv:1203.2889
- Bibcode:
- 2012arXiv1203.2889M
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Number Theory
- E-Print:
- Some minor edits made