Splitting Brauer classes by genus one curves over number fields
arXiv:2607.26915
summary
The authors show that every Brauer class over a number field can be split by a genus‑one curve by proving that any Severi‑Brauer variety of dimension ≥2 contains a twisted elliptic normal curve.
Abstract
We prove that every Severi-Brauer variety of dimension at least two over a number field contains a twisted elliptic normal curve. Consequently, every Brauer class over a number field is split by a genus one curve. We prove this using the fibration method, by showing that the rational points on a smooth compactification of the Hilbert scheme of twisted elliptic normal curves are dense in its Brauer-Manin set.
29 pages, comments welcome!
Topics & keywords
#brauer groups#severi-brauer varieties#genus one curves#elliptic curves#hilbert scheme#brauer-manin obstructionBrauer classtwisted elliptic normal curvefibration methodHilbert schemeBrauer-Manin setnumber field