On Birch and Swinnerton-Dyer's cubic 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.