Boundedness for finite subgroups of linear algebraic groups
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- by Constantin Shramov and Vadim Vologodsky PDF
- Trans. Amer. Math. Soc. 374 (2021), 9029-9046 Request permission
Abstract:
We show the boundedness of finite subgroups in any anisotropic reductive group over a perfect field that contains all roots of $1$. Also, we provide explicit bounds for orders of finite subgroups of automorphism groups of Severi–Brauer varieties and quadrics over such fields.References
- Cahit Arf, Untersuchungen über quadratische Formen in Körpern der Charakteristik 2. I, J. Reine Angew. Math. 183 (1941), 148–167 (German). MR 8069, DOI 10.1515/crll.1941.183.148
- M. Artin, Brauer-Severi varieties, Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., vol. 917, Springer, Berlin-New York, 1982, pp. 194–210. MR 657430
- Roman Bezrukavnikov, Ivan Mirković, and Dmitriy Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. of Math. (2) 167 (2008), no. 3, 945–991. With an appendix by Bezrukavnikov and Simon Riche. MR 2415389, DOI 10.4007/annals.2008.167.945
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- N. Bourbaki, Éléments de mathématique. 23. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1261, Hermann, Paris, 1958 (French). MR 0098114
- A. Borel and J. Tits, Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math. 12 (1971), 95–104 (French). MR 294349, DOI 10.1007/BF01404653
- Tatiana Bandman and Yuri G. Zarhin, Jordan groups, conic bundles and abelian varieties, Algebr. Geom. 4 (2017), no. 2, 229–246. MR 3620637, DOI 10.14231/AG-2017-012
- J. Cassels and A. Fröhlich (eds), Algebraic number theory, Academic Press, 1967.
- François Châtelet, Variations sur un thème de H. Poincaré, Ann. Sci. École Norm. Sup. (3) 61 (1944), 249–300 (French). MR 0014720
- Marcel Herzog and Cheryl E. Praeger, On the order of linear groups of fixed finite exponent, J. Algebra 43 (1976), no. 1, 216–220. MR 424960, DOI 10.1016/0021-8693(76)90156-3
- Mario Garcia-Armas, Finite group actions on curves of genus zero, J. Algebra 394 (2013), 173–181. MR 3092716, DOI 10.1016/j.jalgebra.2013.07.018
- Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528, DOI 10.1017/CBO9780511607219
- Max Lieblich, Twisted sheaves and the period-index problem, Compos. Math. 144 (2008), no. 1, 1–31. MR 2388554, DOI 10.1112/S0010437X07003144
- Hermann Minkowski, Zur Theorie der positiven quadratischen Formen, J. Reine Angew. Math. 101 (1887), 196–202 (German). MR 1580123, DOI 10.1515/crll.1887.101.196
- James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
- Jean-Pierre Serre, Bounds for the orders of the finite subgroups of $G(k)$, Group representation theory, EPFL Press, Lausanne, 2007, pp. 405–450. MR 2336645
- Schémas en groupes. III: Structure des schémas en groupes réductifs, Lecture Notes in Mathematics, Vol. 153, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274460
- T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4
- Ken ichi Tahara, On the finite subgroups of $\textrm {GL}(3,\,Z)$, Nagoya Math. J. 41 (1971), 169–209. MR 272910
- Jacques Tits, Unipotent elements and parabolic subgroups of reductive groups. II, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 265–284. MR 911145, DOI 10.1007/BFb0079243
- J. H. M. Wedderburn, On division algebras, Trans. Amer. Math. Soc. 22 (1921), no. 2, 129–135. MR 1501164, DOI 10.1090/S0002-9947-1921-1501164-3
Additional Information
- Constantin Shramov
- Affiliation: Steklov Mathematical Institute of RAS, 8 Gubkina street, Moscow 119991, Russia; and National Research University Higher School of Economics, Laboratory of Algebraic Geometry, NRU HSE, 6 Usacheva str., Moscow 117312, Russia
- MR Author ID: 907948
- Email: costya.shramov@gmail.com
- Vadim Vologodsky
- Affiliation: National Research University Higher School of Economics, Laboratory of Mirror Symmetry, NRU HSE, 6 Usacheva str., Moscow 117312, Russia
- Email: vologod@gmail.com
- Received by editor(s): March 26, 2021
- Received by editor(s) in revised form: June 3, 2021, and June 22, 2021
- Published electronically: September 29, 2021
- Additional Notes: The first author was partially supported by the Russian Academic Excellence Project “5-100”, by Young Russian Mathematics award, and by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”
The second author was partially supported by the Laboratory of Mirror Symmetry NRU HSE, RF government grant, ag. No. 14.641.31.0001. - © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 9029-9046
- MSC (2020): Primary 20G15, 14J50
- DOI: https://doi.org/10.1090/tran/8511
- MathSciNet review: 4337937