Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T07:52:28.217Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

Artin, M., Algebraic approximation of structures over complete local rings , Publ. Math. Inst. Hautes Études Sci. 36 (1969), 23–58.Google Scholar
Berkovich, V. G., Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33 (American Mathematical Society, Providence, RI, 1990).Google Scholar
Berkovich, V. G., Étale cohomology for non-Archimedean analytic spaces , Publ. Math. Inst. Hautes Études Sci. 78 (1993), 5–161.Google Scholar
Berkovich, V. G., Vanishing cycles for formal schemes , Invent. Math. 115 (1994), 539–571.Google Scholar
Berkovich, V. G., Vanishing cycles for formal schemes. II , Invent. Math. 125 (1996), 367–390.Google Scholar
Bosch, S., Eine bemerkenswerte Eigenschaft der formellen Fasern affinoider Räume , Math. Ann. 229 (1977), 25–45 (in German).Google Scholar
Bosch, S., Güntzer, U. and Remmert, R., Non-Archimedean analysis: a systematic approach to rigid analytic geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261 (Springer, Berlin, 1984).Google Scholar
Bourbaki, N., Éléments de mathématique. Espaces vectoriels topologiques (Hermann, Paris, 1953), ch. 1–5 (in French).Google Scholar
Colliot-Thélène, J.-L., Parimala, R. and Suresh, V., Patching and local–global principles for homogeneous spaces over function fields of p-adic curves , Comment. Math. Helv. 87 (2012), 1011–1033.Google Scholar
Ducros, A., La structure des courbes analytiques, https://webusers.imj-prg.fr/∼antoine.ducros/trirss.pdf.Google Scholar
Ducros, A., Les espaces de Berkovich sont excellents , Ann. Inst. Fourier (Grenoble) 59 (2009), 1443–1552 (in French, with English and French summaries).Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II , Publ. Math. Inst. Hautes Études Sci. 24 (1965), (in French).Google Scholar
Harbater, D., Galois coverings of the arithmetic line , in Number theory, Lecture Notes in Mathematics, vol. 1240 (Springer, Berlin, 1987), 165–195.Google Scholar
Harbater, D., Galois covers of an arithmetic surface , Amer. J. Math. 110 (1988), 849–885.Google Scholar
Harbater, D., Patching and Galois theory , in Galois groups and fundamental groups, Mathematical Sciences Research Institute Publications, vol. 41 (Cambridge University Press, Cambridge, 2003), 313–424.Google Scholar
Harbater, D. and Hartmann, J., Patching over fields , Israel J. Math. 176 (2010), 61–107.Google Scholar
Harbater, D., Hartmann, J. and Krashen, D., Applications of patching to quadratic forms and central simple algebras , Invent. Math. 178 (2009), 231–263.Google Scholar
Harbater, D., Hartmann, J. and Krashen, D., Local–global principles for torsors over arithmetic curves , Amer. J. Math. 137 (2015), 1559–1612.Google Scholar
Leep, D. B., The u-invariant of p-adic function fields , J. reine angew. Math. 679 (2013), 65–73.Google Scholar
Liu, Q., Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6 (Oxford University Press, Oxford, 2002). Translated from the French by Reinie Erné, Oxford Science Publications.Google Scholar
Marker, D., Model theory: an introduction, Graduate Texts in Mathematics, vol. 217 (Springer, New York, 2002).Google Scholar
Martin, F., Analytic functions on tubes of nonarchimedean analytic spaces , Algebra Number Theory 11 (2017), 657–683. With an appendix by Christian Kappen and Martin.Google Scholar
Milne, J. S., On the arithmetic of abelian varieties , Invent. Math. 17 (1972), 177–190.Google Scholar
Mustaţă, M. and Nicaise, J., Weight functions on non-Archimedean analytic spaces and the Kontsevich–Soibelman skeleton , Algebr. Geom. 2 (2015), 365–404.Google Scholar
Parimala, R. and Suresh, V., The u-invariant of the function fields of p-adic curves , Ann. of Math. (2) 172 (2010), 1391–1405.Google Scholar
Poineau, J., Raccord sur les espaces de Berkovich , Algebra Number Theory 4 (2010), 297–334 (in French, with English and French summaries).Google Scholar
Temkin, M., On local properties of non-Archimedean analytic spaces. II , Israel J. Math. 140 (2004), 1–27.Google Scholar
Temkin, M., A new proof of the Gerritzen–Grauert theorem , Math. Ann. 333 (2005), 261–269.Google Scholar
Thuillier, A., Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov, Thèse de l’Université de Rennes 1 (2005).Google Scholar
Vázquez, R. R., Non-Archimedean normal families, Preprint (2016), arXiv:1607.05976.pdf.Google Scholar