Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Transcendental Brauer groups of products of CM elliptic curves

Rachel Newton

Abstract

Let $L$ be a number field and let $E/L$ be an elliptic curve with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$. We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface $E\times E$. The results for the odd order torsion also apply to the Brauer group of the K3 surface $\textrm{Kum}(E\times E)$. We describe explicitly the elliptic curves $E/\mathbb{Q}$ with complex multiplication by $\mathcal{O}_K$ such that the Brauer group of $E\times E$ contains a transcendental element of odd order. We show that such an element gives rise to a Brauer-Manin obstruction to weak approximation on $\textrm{Kum}(E\times E)$, while there is no obstruction coming from the algebraic part of the Brauer group.

Transcendental Brauer groups of products of CM elliptic curves

Abstract

Let be a number field and let be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field . We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface . The results for the odd order torsion also apply to the Brauer group of the K3 surface . We describe explicitly the elliptic curves with complex multiplication by such that the Brauer group of contains a transcendental element of odd order. We show that such an element gives rise to a Brauer-Manin obstruction to weak approximation on , while there is no obstruction coming from the algebraic part of the Brauer group.

Paper Structure

This paper contains 13 sections, 24 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Transcendental Brauer group computations
  4. Preliminaries
  5. Case I: Complex multiplication defined over the base field.
  6. Case II: Complex multiplication not defined over the base field.
  7. Special cases and examples
  8. Computing $m(2)$
  9. Computing $m(3)$
  10. The Brauer group of $\mathop{\mathrm{Kum}}\nolimits(E\times E)$
  11. A transcendental Brauer-Manin obstruction to weak approximation
  12. Errata
  13. Corrections

Key Result

Theorem \oldthetheorem

Let $\ell\in{\mathbb Z}_{>0}$ be an odd prime and let $m=m(\ell)$. Then

Theorems & Definitions (52)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • ...and 42 more