On an isotropy criterion for quadratic forms over function fields of curves over non-dyadic complete discrete valuation rings

Tom 108 / 2016

David Grimm Banach Center Publications 108 (2016), 95-103 MSC: 11G99, 14G99, 14G05, 11E12, 20G35. DOI: 10.4064/bc108-0-8


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.


  • David GrimmDepartamento de Matemáticas
    Facultad de Ciencia
    Universidad de Santiago de Chile
    Av. Libertador Bernardo O’Higgins 3363
    Santiago, Chile

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