Explicit bounds on the transcendental Brauer group of K3 surfaces with principal complex multiplication

TL;DR

The paper provides explicit, computable bounds on the size of the transcendental Brauer group for K3 surfaces with principal complex multiplication by a CM field $E$, over a number field $k$. Building on Valloni’s CM-arithmetic framework, it derives a crude bound and a sharpened bound using the totient structure and new auxiliary functions $\\Phi$ and $\\Psi$ to control products over primes; it also analyzes imaginary-quadratic CM to obtain tight asymptotics. Central to the results is bounding $\\mathrm{Nm}(I)$ for $k$-permissible ideals $I$ via explicit control of unit-index ratios, ramification, and Tate cohomology, culminating in an explicit bound that scales with $[k: olinebreak\mathbb{Q}]$, $[E:\mathbb{Q}]$, and arithmetic invariants of $E$ such as $d_{E/F}$. The practical impact lies in enabling effective computation of the Brauer--Manin obstruction for CM K3 surfaces and clarifying how CM-arithmetic data governs the transcendental Brauer group size.

Abstract

Let $X$ be a K3 surface defined over a number field $k$, with principal complex multiplication by a CM field $E$. We find explicit bounds, in terms of $k$ and $E$, on the size of the transcendental Brauer group $\operatorname{Br}(X)/\operatorname{Br}_1(X)$ of $X$. Bounding the size of this group is important for computing the Brauer--Manin obstruction, which is conjectured by Skorobogatov to be the only obstruction to the Hasse principle for K3 surfaces. Our methods are built on top of earlier work by Valloni, who related the group $\operatorname{Br}(X)/\operatorname{Br}_1(X)$ to the arithmetic structure of the CM field $E$. It is from this arithmetic structure that we deduce our bounds.

Theorems & Definitions (59)

• Conjecture 1.1: Skorobogatov
• Theorem 1.2
• Theorem 1.3
• Example 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Example 1.8: Imaginary quadratic fields
• Definition 2.1
• Theorem 2.2