[Un point de vue analytique sur le principe de Hasse]
En travaillant sur des courbes analytiques non archimédiennes, nous proposons une approche géométrique de l’étude du principe de Hasse sur les corps de fonctions des courbes définies sur un corps discrètement valué complet. En l’utilisant, nous montrons que le principe de Hasse est vérifié pour certaines familles d’espaces projectifs homogènes. En conséquence, nous prouvons que ce principe est valable pour les formes quadratiques et les variétés homogènes sur les groupes unitaires, résultats initialement démontrés dans [8], [36], et [33].
Working on non-Archimedean analytic curves, we propose a geometric approach to the study of the Hasse principle over function fields of curves defined over a complete discretely valued field. Using it, we show the Hasse principle to be verified for certain families of projective homogeneous spaces. As a consequence, we prove that said principle holds for quadratic forms and homogeneous varieties over unitary groups, results originally shown in [8], [36], and [33].
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Keywords: Local-global principle, Hasse principle, Berkovich curve, discrete valuation, homogeneous space, quadratic form
Mot clés : Principe local-global, principe de Hasse, courbe de Berkovich, valuation discrète, espace homogène, forme quadratique
Vlerë Mehmeti 1
CC-BY 4.0
@article{JEP_2024__11__875_0,
author = {Vler\"e Mehmeti},
title = {An analytic viewpoint on the {Hasse} principle},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {875--908},
publisher = {\'Ecole polytechnique},
volume = {11},
year = {2024},
doi = {10.5802/jep.268},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.268/}
}
TY - JOUR AU - Vlerë Mehmeti TI - An analytic viewpoint on the Hasse principle JO - Journal de l’École polytechnique — Mathématiques PY - 2024 SP - 875 EP - 908 VL - 11 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.268/ DO - 10.5802/jep.268 LA - en ID - JEP_2024__11__875_0 ER -
Vlerë Mehmeti. An analytic viewpoint on the Hasse principle. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 875-908. doi : 10.5802/jep.268. https://jep.centre-mersenne.org/articles/10.5802/jep.268/
[1] - Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990
[2] - “Vanishing cycles for formal schemes”, Invent. Math. 115 (1994) no. 3, p. 539-571 | DOI | Zbl
[3] - “Eine bemerkenswerte Eigenschaft der formellen Fasern affinoider Räume”, Math. Ann. 229 (1977) no. 1, p. 25-45 | DOI | Zbl
[4] - “The Hasse principle for groups of type E8”, Dokl. Akad. Nauk SSSR 306 (1989) no. 5, p. 1059-1063
[5] - “R-equivalence and special unitary groups”, J. Algebra 209 (1998) no. 1, p. 175-198 | DOI | Zbl
[6] - “The rationality problem for semisimple group varieties”, J. reine angew. Math. 504 (1998), p. 1-28 | Zbl
[7] - “Local-global principles for constant reductive groups over semi-global fields”, Michigan Math. J. 72 (2022), p. 77-144 | Zbl
[8] - “Patching and local-global principles for homogeneous spaces over function fields of -adic curves”, Comment. Math. Helv. 87 (2012) no. 4, p. 1011-1033 | DOI | Zbl
[9] - “La structure des courbes analytiques”, https://webusers.imj-prg.fr/~antoine.ducros/trirss.pdf
[10] - “Triangulations of non-Archimedean curves, semi-stable reduction, and ramification”, Ann. Inst. Fourier (Grenoble) 73 (2023) no. 2, p. 695-746 | DOI | Zbl
[11] - “The index of an algebraic variety”, Invent. Math. 192 (2013) no. 3, p. 567-626 | DOI | Zbl
[12] - “Springer’s odd degree extension theorem for quadratic forms over semilocal rings”, Indag. Math. (N.S.) 32 (2021) no. 6, p. 1290-1310 | DOI | Zbl
[13] - “Une propriété des couples henséliens”, in Colloque d’algèbre commutative (Rennes, 1972), 1972, Exp. No. 10, 13 p. | Numdam | Zbl
[14] - “Local-global principle for hermitian spaces over semi-global fields”, 2022 | arXiv
[15] - “Applications of patching to quadratic forms and central simple algebras”, Invent. Math. 178 (2009) no. 2, p. 231-263 | DOI | Zbl
[16] - “Refinements to patching and applications to field invariants”, Internat. Math. Res. Notices (2015) no. 20, p. 10399-10450 | DOI | Zbl
[17] - “Über die Galoiskohomologie halbeinfacher Matrizengruppen. I”, Math. Z. 90 (1965), p. 404-428 | DOI | Zbl
[18] - “Über die Galoiskohomologie halbeinfacher Matrizengruppen. II”, Math. Z. 92 (1966), p. 396-415 | DOI | Zbl
[19] - “Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III”, J. reine angew. Math. 274/275 (1975), p. 125-138 | Zbl
[20] - “Crystalline Dieudonné module theory via formal and rigid geometry”, Publ. Math. Inst. Hautes Études Sci. 82 (1995), p. 5-96 | DOI | Numdam | Zbl
[21] - “Galois-Kohomologie halbeinfacher algebraischer Gruppen über -adischen Körpern. I”, Math. Z. 88 (1965), p. 40-47 | DOI | Zbl
[22] - “Galois-Kohomologie halbeinfacher algebraischer Gruppen über -adischen Körpern. II”, Math. Z. 89 (1965), p. 250-272 | DOI
[23] - The book of involutions, Amer. Math. Soc. Colloq. Publ., vol. 44, American Mathematical Society, Providence, RI, 1998
[24] - Introduction to quadratic forms over fields, Graduate Studies in Math., vol. 67, American Mathematical Society, Providence, RI, 2005
[25] - “A Springer theorem for Hermitian forms”, Math. Z. 252 (2006) no. 3, p. 459-472 | DOI | Zbl
[26] - Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Math., vol. 6, Oxford University Press, Oxford, 2002 | DOI
[27] - “Analytic functions on tubes of nonarchimedean analytic spaces”, Algebra Number Theory 11 (2017) no. 3, p. 657-683 | DOI | Zbl
[28] - Patching on Berkovich spaces and the local–global principle, Ph. D. Thesis, University of Caen Normandy, 2019
[29] - “Patching over Berkovich curves and quadratic forms”, Compositio Math. 155 (2019) no. 12, p. 2399-2438 | DOI | Zbl
[30] - “Weight functions on non-Archimedean analytic spaces and the Kontsevich-Soibelman skeleton”, Algebraic Geom. 2 (2015) no. 3, p. 365-404 | DOI | Zbl
[31] - “Homogeneous varieties—zero-cycles of degree one versus rational points”, Asian J. Math. 9 (2005) no. 2, p. 251-256 | DOI | Zbl
[32] - “Local-global principle for reduced norms over function fields of -adic curves”, Compositio Math. 154 (2018) no. 2, p. 410-458 | DOI | Zbl
[33] - “Local-global principle for classical groups over function fields of -adic curves”, Comment. Math. Helv. 97 (2022), p. 255-304 | DOI | Zbl
[34] - “Admissibility of groups over function fields of -adic curves”, Adv. Math. 237 (2013), p. 316-330 | DOI | Zbl
[35] - “The Stacks Project”, 2019, https://stacks.math.columbia.edu
[36] - “Hasse principle for hermitian spaces over semi-global fields”, J. Algebra 458 (2016), p. 171-196 | DOI | Zbl
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