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Boundedness of the p-primary torsion of the Brauer groups of K3 surfaces

Christopher D. Lazda, Alexei N. Skorobogatov

TL;DR

The paper proves that for a K3 surface X over a finitely generated field k, the p-primary part of the transcendental Brauer group Br(bar(X))^k is finite unless k has characteristic p>0 and X is supersingular, in which case it is annihilated by p. It achieves this by combining crystalline cohomology with the Kuga-Satake construction: for non-supersingular X there is a horizontal Frobenius-compatible embedding of the primitive crystalline cohomology into endomorphisms of a Kuga-Satake abelian variety A, linking the p-adic Brauer group to special endomorphisms of A. The argument reduces the transcendental Brauer group problem to the crystalline Tate conjecture for abelian varieties (proved by de Jong) and to the Tate conjecture for K3 surfaces (now known), yielding explicit p-torsion formulas in finite-height cases and annihilation by p in the supersingular case. By spreading out in families and leveraging these correspondences, the authors establish a precise boundedness statement for the p-primary torsion on Br(barX)^k and discuss corollaries over infinite finitely generated fields as well as potential extensions.

Abstract

We prove that the transcendental Brauer group of a K3 surface X over a finitely generated field k is finite, unless k has positive characteristic p and X is supersingular, in which case it is annihilated by p.

Boundedness of the p-primary torsion of the Brauer groups of K3 surfaces

TL;DR

The paper proves that for a K3 surface X over a finitely generated field k, the p-primary part of the transcendental Brauer group Br(bar(X))^k is finite unless k has characteristic p>0 and X is supersingular, in which case it is annihilated by p. It achieves this by combining crystalline cohomology with the Kuga-Satake construction: for non-supersingular X there is a horizontal Frobenius-compatible embedding of the primitive crystalline cohomology into endomorphisms of a Kuga-Satake abelian variety A, linking the p-adic Brauer group to special endomorphisms of A. The argument reduces the transcendental Brauer group problem to the crystalline Tate conjecture for abelian varieties (proved by de Jong) and to the Tate conjecture for K3 surfaces (now known), yielding explicit p-torsion formulas in finite-height cases and annihilation by p in the supersingular case. By spreading out in families and leveraging these correspondences, the authors establish a precise boundedness statement for the p-primary torsion on Br(barX)^k and discuss corollaries over infinite finitely generated fields as well as potential extensions.

Abstract

We prove that the transcendental Brauer group of a K3 surface X over a finitely generated field k is finite, unless k has positive characteristic p and X is supersingular, in which case it is annihilated by p.
Paper Structure (4 sections, 10 theorems, 73 equations)

This paper contains 4 sections, 10 theorems, 73 equations.

Table of Contents

  1. The geometric Brauer group
  2. Crystalline cohomology
  3. The Kuga--Satake construction
  4. Completion of the proof

Key Result

Theorem A

Suppose that $k$ is finitely generated of characteristic $p>0$, and that $X$ is a K3 surface. In particular, ${\rm{Br}}(\overline{X})^k\{p\}$ is of finite exponent.

Theorems & Definitions (21)

  • Theorem A
  • Corollary B
  • Corollary C
  • Theorem 1.1
  • proof
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • ...and 11 more