Harbater, Hartmann and Krashen obtained in 2015 a criterion for the existence of rational points on projective (or principal) homogeneous varieties for rational connected algebraic groups defined over function fields of normal curves over a complete discrete valuation ring in terms of completions of local rings at special points. This was obtained by a reduction via Artin approximation to a related patching problem solved by the same authors in 2009. In the special case of projective quadrics, we present a more elementary reduction in the non-dyadic case. The proof is strongly inspired by the proof of a more Hasse-like local-global principle due to Colliot-Thélène, Parimala and Suresh, and we present a variant of their proof based on the mentioned criterion.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc108-0-8,
author = {David Grimm},
title = {On an isotropy criterion for quadratic forms over function fields of curves over non-dyadic complete discrete valuation rings},
journal = {Banach Center Publications},
volume = {108},
year = {2016},
pages = {95-103},
zbl = {06622288},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc108-0-8}
}
David Grimm. On an isotropy criterion for quadratic forms over function fields of curves over non-dyadic complete discrete valuation rings. Banach Center Publications, Tome 108 (2016) pp. 95-103. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc108-0-8/