A remark on uniform boundedness for Brauer groups

TL;DR

This paper demonstrates that the finiteness of the $ ext{ell}$-primary torsion in the Brauer group, linked to the Tate conjecture, is uniformly bounded in one-dimensional families of certain varieties like abelian and K3 surfaces.

Contribution

It proves the uniform boundedness of $ ext{ell}$-primary torsion in the Brauer group for families satisfying the Tate conjecture, extending previous finiteness results.

Findings

Finiteness of $ ext{ell}$-primary torsion is uniform in families.

Applicable to abelian varieties and K3 surfaces.

Supports the Tate conjecture's implications for Brauer groups.

Abstract

The Tate conjecture for divisors on varieties over number fields is equivalent to finiteness of $\ell$-primary torsion in the Brauer group. We show that this finiteness is actually uniform in one-dimensional families for varieties that satisfy the Tate conjecture for divisors -- e.g. abelian varieties and $K3$ surfaces.