On Birch and Swinnerton-Dyer's cubic surfaces

TL;DR

This paper extends the work of Birch and Swinnerton-Dyer by using computer algebra to demonstrate that many cubic surfaces exhibit Brauer--Manin obstructions to the Hasse principle, supporting a conjecture in algebraic geometry.

Contribution

It generalizes previous results by identifying a broader class of cubic surfaces with Brauer--Manin obstructions, verifying the Colliot-Thélène--Sansuc conjecture for infinitely many cases.

Findings

Many cubic surfaces have a Brauer--Manin obstruction to the Hasse principle.

Verification of the Colliot-Thélène--Sansuc conjecture for infinitely many cubic surfaces.

Use of modern computer algebra software to analyze algebraic surfaces.

Abstract

In a 1975 paper of Birch and Swinnerton-Dyer, a number of explicit norm form cubic surfaces are shown to fail the Hasse Principle. They make a correspondence between this failure and the Brauer--Manin obstruction, recently discovered by Manin. We generalize their work, making use of modern computer algebra software to show that a larger set of cubic surfaces have a Brauer--Manin obstruction to the Hasse principle, thus verifying the Colliot-Th\'el\`ene--Sansuc conjecture for infinitely many cubic surfaces.