Given a surface over a field of characteristic , Artin conjectured that if is supersingular (meaning infinite height), then its Picard rank is 22. Along with work of Nygaard–Ogus, this conjecture implies the Tate conjecture for surfaces over finite fields with . We prove Artin’s conjecture under the additional assumption that has a polarization of degree with . Assuming semistable reduction for surfaces in characteristic , we can improve the main result to surfaces which admit a polarization of degree prime to when .
The argument uses Borcherds’s construction of automorphic forms on 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 degenerations. In the appendix by A. Snowden, a compatibility statement is proven between Clifford constructions and integral -adic comparison functors.
"Supersingular K3 surfaces for large primes." Duke Math. J. 163 (13) 2357 - 2425, 1 October 2014. https://doi.org/10.1215/00127094-2804783