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Patching over Berkovich curves and quadratic forms

Part of: Arithmetic problems. Diophantine geometry Forms and linear algebraic groups

Published online by Cambridge University Press:  11 November 2019

Vlerë Mehmeti
Vlerë Mehmeti*
Affiliation:
Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Normandie, 14032 Caen Cedex, France email vlere.mehmeti@unicaen.fr
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Abstract

We extend field patching to the setting of Berkovich analytic geometry and use it to prove a local–global principle over function fields of analytic curves with respect to completions. In the context of quadratic forms, we combine it with sufficient conditions for local isotropy over a Berkovich curve to obtain applications on the $u$-invariant. The patching method we adapt was introduced by Harbater and Hartmann [Patching over fields, Israel J. Math. 176 (2010), 61–107] and further developed by these two authors and Krashen [Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), 231–263]. The results presented in this paper generalize those of Harbater, Hartmann, and Krashen [Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), 231–263] on the local–global principle and quadratic forms.

Keywords

Berkovich analytic curveslocal–global principlequadratic formsu-invariant

MSC classification

Primary: 14G22: Rigid analytic geometry 11E08: Quadratic forms over local rings and fields
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 12 , December 2019 , pp. 2399 - 2438
Copyright
© The Author 2019 

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Footnotes

The author was supported by the ERC Starting Grant ‘TOSSIBERG’: 637027.

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