Transcendental Brauer groups of cubic generalised Kummer surfaces

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