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Transcendental Brauer-Manin obstructions on singular K3 surfaces

Mohamed Alaa Tawfik, Rachel Newton

TL;DR

The work studies transcendental Brauer–Manin obstructions on singular K3 surfaces $Y=\mathrm{Kum}(E\times E')$ arising from CM elliptic curves, identifying precise CM-field cases and order bounds for odd-order Brauer classes. It develops a framework that connects $\mathrm{Br}(Y)$ to abelian-variety Brauer groups and uses cup-product evaluations to compute local obstructions, yielding new weak-approximation obstructions unattached to algebraic Brauer elements. For CM by ordinary imaginary quadratic fields with $\mathcal{O}_K^{\times}=\{\pm1\}$, the transcendental part is tightly constrained (exponent divides $6$) with explicit $K$-dependent phenomena; in the $\mathbb{Z}[\zeta_3]$-CM case, odd-order elements of orders up to $9$ appear under sextic reciprocity conditions. The paper also proves that many evaluation maps are constant away from the relevant primes, while establishing surjectivity in split cases and handling inert cases via Harpaz–Skorobogatov, thereby constructing concrete Brauer–Manin obstructions to weak approximation on these Kummer surfaces.

Abstract

Let E and E' be elliptic curves over Q with complex multiplication by the ring of integers of an imaginary quadratic field K and let Y=Kum(ExE') be the minimal desingularisation of the quotient of ExE' by the action of -1. We study the Brauer groups of such surfaces Y and use them to furnish new examples of transcendental Brauer-Manin obstructions to weak approximation.

Transcendental Brauer-Manin obstructions on singular K3 surfaces

TL;DR

The work studies transcendental Brauer–Manin obstructions on singular K3 surfaces arising from CM elliptic curves, identifying precise CM-field cases and order bounds for odd-order Brauer classes. It develops a framework that connects to abelian-variety Brauer groups and uses cup-product evaluations to compute local obstructions, yielding new weak-approximation obstructions unattached to algebraic Brauer elements. For CM by ordinary imaginary quadratic fields with , the transcendental part is tightly constrained (exponent divides ) with explicit -dependent phenomena; in the -CM case, odd-order elements of orders up to appear under sextic reciprocity conditions. The paper also proves that many evaluation maps are constant away from the relevant primes, while establishing surjectivity in split cases and handling inert cases via Harpaz–Skorobogatov, thereby constructing concrete Brauer–Manin obstructions to weak approximation on these Kummer surfaces.

Abstract

Let E and E' be elliptic curves over Q with complex multiplication by the ring of integers of an imaginary quadratic field K and let Y=Kum(ExE') be the minimal desingularisation of the quotient of ExE' by the action of -1. We study the Brauer groups of such surfaces Y and use them to furnish new examples of transcendental Brauer-Manin obstructions to weak approximation.
Paper Structure (14 sections, 39 theorems, 122 equations, 1 table)

This paper contains 14 sections, 39 theorems, 122 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Transcendental Brauer groups
  5. Evaluation maps
  6. CM by ${\mathcal{O}}_K$ with ${\mathcal{O}}_K^\times=\{\pm 1\}$: transcendental Brauer groups
  7. $\bullet\ K={\mathbb Q}(\sqrt{-2})$:
  8. $\bullet\ K={\mathbb Q}(\sqrt{-11})$:
  9. CM by ${\mathbb Z}[\zeta_3]$: transcendental Brauer groups
  10. Algebraic Brauer groups
  11. Constant evaluation maps
  12. Surjective evaluation maps
  13. The case of $\ell$ split in $K/{\mathbb Q}$
  14. The case of $\ell$ inert in $K/{\mathbb Q}$

Key Result

Theorem 1.1

Let $K$ be an imaginary quadratic field and let $Y=\mathop{\mathrm{Kum}}\nolimits(E\times E')$ for elliptic curves $E,E'$ over ${\mathbb Q}$ with $\mathop{\mathrm{End}}\nolimits \bar{E}=\mathop{\mathrm{End}}\nolimits\bar{E}'={\mathcal{O}}_K$. Suppose that $\mathop{\mathrm{Br}}\nolimits(Y)/\mathop{\m

Theorems & Definitions (83)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 2.1: SZtorsion
  • Proposition 2.2: SZtorsion
  • Definition 2.3
  • ...and 73 more