Supersingular K3 surfaces for large primes

TL;DR

This paper proves Artin's conjecture relating supersingular K3 surfaces' Picard rank to their height in characteristic p, using automorphic forms and moduli space techniques, with implications for the Tate conjecture.

Contribution

It establishes Artin's conjecture for supersingular K3 surfaces under certain polarization and prime conditions, advancing understanding of their geometric and arithmetic properties.

Findings

Proves Artin's conjecture for supersingular K3 surfaces with specific polarizations.

Constructs ample divisors on moduli spaces using automorphic forms.

Establishes positivity results and classification of K3 degenerations in characteristic p.

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.