Persistence of the Brauer-Manin obstruction on cubic surfaces

TL;DR

The paper proves that the Brauer-Manin obstruction to rational points on cubic surfaces remains persistent over certain extensions, linking it to conjectures about rational points and zero-cycles.

Contribution

It establishes the invariance of the Brauer-Manin obstruction under extensions of degree prime to 3 for cubic surfaces, connecting it to broader conjectures.

Findings

Persistence of the Brauer-Manin obstruction over extensions with degree coprime to 3.

Equivalence between the existence of rational points and zero-cycles of degree 1 under the conjecture.

Implication that the conjecture of Colliot-Thélène and Sansuc implies a special case of Cassels and Swinnerton-Dyer.

Abstract

Let $X$ be a cubic surface over a global field $k$. We prove that a Brauer-Manin obstruction to the existence of $k$-points on $X$ will persist over every extension $L/k$ with degree relatively prime to $3$. In other words, a cubic surface has nonempty Brauer set over $k$ if and only if it has nonempty Brauer set over some extension $L/k$ with $3\nmid[L:k]$. Therefore, the conjecture of Colliot-Th\'el\`ene and Sansuc on the sufficiency of the Brauer-Manin obstruction for cubic surfaces implies that $X$ has a $k$-rational point if and only if $X$ has a $0$-cycle of degree $1$. This latter statement is a special case of a conjecture of Cassels and Swinnerton-Dyer.