ScholarLens Back to search
1993•Mathematical Proceedings of the Cambridge Philosophical SocietyOpen access

The Brauer group of cubic surfaces

Sir Peter Swinnerton-Dyer

Open full text 49 citations

Abstract

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injection is an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

Open-access reader

About this research paper

What this paper is about

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injection is an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

Why it matters

OpenAlex reports 49 citations for this work. Citation counts describe recorded attention and do not establish research quality.

Key contribution

A contribution statement is not available in the OpenAlex record.

Method / approach

Method details are not available in the OpenAlex metadata.

Main findings

Findings are not separately available in the OpenAlex metadata.

Limitations

Limitations are not available in the OpenAlex metadata.

Applications

Application details are not available in the OpenAlex metadata.

Available abstract

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injection is an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

Key concepts: Brauer group, Mathematics, Hasse principle, Isomorphism (crystallography), Group (periodic table), Pure mathematics, Algebraic group, Brauer's theorem on induced characters

Related papers

Related paper 1Related paper 2Related paper 3
Back to paper searchBrowse research topicsOriginal source
The Brauer group of cubic surfaces — Research Paper | ScholarLens