Local-global principles for tori over arithmetic curves
Algebraic Geometry
2020-11-24 v3 Number Theory
Abstract
In this paper we study local-global principles for tori over semi-global fields, which are one variable function fields over complete discretely valued fields. In particular, we show that for principal homogeneous spaces for tori over the underlying discrete valuation ring, the obstruction to a local-global principle with respect to discrete valuations can be computed using methods coming from patching. We give a sufficient condition for the vanishing of the obstruction, as well as examples were the obstruction is nontrivial or even infinite. A major tool is the notion of a flasque resolution of a torus.
Cite
@article{arxiv.1906.10672,
title = {Local-global principles for tori over arithmetic curves},
author = {Jean-Louis Colliot-Thélène and David Harbater and Julia Hartmann and Daniel Krashen and R. Parimala and V. Suresh},
journal= {arXiv preprint arXiv:1906.10672},
year = {2020}
}
Comments
27 pages; final pre-publication version; correction to Theorem 6.5, rewording at Proposition 6.3 and beginning of Section 8.3