Transcendental Brauer groups of cubic generalised Kummer surfaces

TL;DR

This paper investigates the transcendental Brauer groups of certain K3 surfaces derived from cubic curves, providing explicit computations and conjectural insights into rational points over number fields.

Contribution

It explicitly computes the transcendental Brauer groups of K3 surfaces associated with diagonal cubic curves over  and relates these to the properties of the original curves.

Findings

Explicit computation of transcendental Brauer groups for diagonal cubic curves over .

Conjectural links between Brauer groups and the existence of Galois cubic points.

Insights into rational points on cubic curves over .

Abstract

Given a cubic curve $C$ over a number field, we consider the K3 surface $Y_C$ constructed as the minimal desingularisation of the quotient of $C \times C$ by an automorphism of order 3. We relate the transcendental Brauer groups of $Y_C$ and $C \times C$, allowing us to explicitly compute the former group in the case of a diagonal cubic curve defined over $\mathbb{Q}$. We obtain conjectural insight on the existence of Galois cubic points over $\mathbb{Q}$ for everywhere locally soluble diagonal cubic curves.