%0 Generic %1 maulik2012supersingular %A Maulik, Davesh %D 2012 %K k3 large supersingular surfaces mathematics %T Supersingular K3 surfaces for large primes %U http://arxiv.org/abs/1203.2889 %X 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' 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.

@misc{maulik2012supersingular, 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.}, added-at = {2013-12-23T06:33:08.000+0100}, author = {Maulik, Davesh}, biburl = {https://www.bibsonomy.org/bibtex/2688575a576508029cdd2df5a752b61c2/aeu_research}, description = {Supersingular K3 surfaces for large primes}, interhash = {b2a55d3610c2068b70d95ea3b3a2efcb}, intrahash = {688575a576508029cdd2df5a752b61c2}, keywords = {k3 large supersingular surfaces mathematics}, note = {cite arxiv:1203.2889Comment: Some minor edits made; German error fixed; comments still welcome}, timestamp = {2013-12-23T08:02:21.000+0100}, title = {Supersingular K3 surfaces for large primes}, url = {http://arxiv.org/abs/1203.2889}, year = 2012 }