PreprintCC BY 4.0arXiv
Dichotomy for the Brauer-Manin obstruction in characteristic p
Abstract
Let X be a smooth, projective, geometrically integral variety over a global function field k of characteristic p. We show that if the unipotent Brauer group of X is zero, and the Picard scheme of X is torsion free, then only finitely many places of k can be potentially relevant to the Brauer--Manin obstruction for X. In the opposite case, almost all places of k are potentially relevant. If X is base changed from a finite field, then in both cases the finite set of exceptional places is empty. We give an example of a supersingular K3 surface over a global function field k of characteristic p>2 such that Br(X) is finite modulo Br(k), answering a question from our previous paper.
Related
BibTeX
@misc{lazda2026dichotomy,
title = {{Dichotomy for the Brauer-Manin obstruction in characteristic p}},
author = {Christopher D. Lazda and Alexei N. Skorobogatov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.14967},
eprint = {2606.14967},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.14967},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dichotomy for the Brauer-Manin obstruction in characteristic p
AU - Christopher D. Lazda
AU - Alexei N. Skorobogatov
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2606.14967
UR - https://arxiv.org/abs/2606.14967
AB - Let X be a smooth, projective, geometrically integral variety over a global function field k of characteristic p. We show that if the unipotent Brauer group of X is zero, and the Picard scheme of X is torsion free, then only finitely many places of k can be potentially relevant to the Brauer--Manin obstruction for X. In the opposite case, almost all places of k are potentially relevant. If X is base changed from a finite field, then in both cases the finite set of exceptional places is empty. We give an example of a supersingular K3 surface over a global function field k of characteristic p>2 such that Br(X) is finite modulo Br(k), answering a question from our previous paper.
ER -