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 $p\ge 5$. 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 $p\ge 5$.

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.