{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:24:31Z","timestamp":1740108271332,"version":"3.37.3"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"3-4","license":[{"start":{"date-parts":[[2021,11,6]],"date-time":"2021-11-06T00:00:00Z","timestamp":1636156800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,11,6]],"date-time":"2021-11-06T00:00:00Z","timestamp":1636156800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Ann."],"published-print":{"date-parts":[[2022,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this article we construct a new motivic measure called the<jats:italic>Jacques Tits motivic measure<\/jats:italic>. As a first main application, we prove that two Severi-Brauer varieties (or, more generally, two twisted Grassmannian varieties), associated to 2-torsion central simple algebras, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that if two Severi-Brauer varieties, associated to central simple algebras of period<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\{3, 4, 5, 6\\}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mo>{<\/mml:mo><mml:mn>3<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>4<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>5<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>6<\/mml:mn><mml:mo>}<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, have the same class in the Grothendieck ring of varieties, then they are necessarily birational to each other. As a second main application, we prove that two quadric hypersurfaces (or, more generally, two involution varieties), associated to quadratic forms of dimension 6 or to quadratic forms of arbitrary dimension defined over a base field<jats:italic>k<\/jats:italic>with<jats:inline-formula><jats:alternatives><jats:tex-math>$$I^3(k)=0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:msup><mml:mi>I<\/mml:mi><mml:mn>3<\/mml:mn><\/mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>=<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that the latter main application also holds for products of quadric hypersurfaces.<\/jats:p>","DOI":"10.1007\/s00208-021-02292-6","type":"journal-article","created":{"date-parts":[[2021,11,6]],"date-time":"2021-11-06T09:02:22Z","timestamp":1636189342000},"page":"1245-1278","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Jacques Tits motivic measure"],"prefix":"10.1007","volume":"382","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8245-6882","authenticated-orcid":false,"given":"Gon\u00e7alo","family":"Tabuada","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,11,6]]},"reference":[{"key":"2292_CR1","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1090\/S0002-9939-1972-0297803-6","volume":"35","author":"A Albert","year":"1972","unstructured":"Albert, A.: Tensor products of quaternion algebras. Proc. AMS 35, 65\u201366 (1972)","journal-title":"Proc. AMS"},{"key":"2292_CR2","doi-asserted-by":"publisher","first-page":"8","DOI":"10.2307\/2007098","volume":"62","author":"S Amitsur","year":"1955","unstructured":"Amitsur, S.: Generic splitting fields of central simple algebras. Ann. Math. 62, 8\u201343 (1955)","journal-title":"Ann. Math."},{"issue":"4","key":"2292_CR3","doi-asserted-by":"publisher","first-page":"1011","DOI":"10.1112\/S0010437X03000617","volume":"140","author":"F Bittner","year":"2004","unstructured":"Bittner, F.: The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140(4), 1011\u20131032 (2004)","journal-title":"Compos. Math."},{"issue":"5","key":"2292_CR4","first-page":"669","volume":"181","author":"A Bondal","year":"1990","unstructured":"Bondal, A., Kapranov, M.: Framed triangulated categories. Mat. Sb. 181(5), 669\u2013683 (1990)","journal-title":"Mat. Sb."},{"key":"2292_CR5","unstructured":"Bondal, A., Orlov, D.: Derived categories of coherent sheaves. Proceedings of the international congress of mathematicians, Vol. II, pp. 47\u201356. Higher Ed. Press, Beijing (2002)"},{"key":"2292_CR6","unstructured":"Bondal, A., Orlov, D.: Semiorthogonal decomposition for algebraic varieties. Available at arXiv:alg-geom\/9506012 (2021)"},{"key":"2292_CR7","unstructured":"Colmez, P., Serre, J.-P.: Correspondance Grothendieck-Serre. Doc. Math. 2 (2001)"},{"key":"2292_CR8","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1007\/BF02566738","volume":"49","author":"R Elman","year":"1974","unstructured":"Elman, R., Lam, T.-Y.: Classification theorems for quadratic forms over fields. Comment. Math. Helv. 49, 373\u2013381 (1974)","journal-title":"Comment. Math. Helv."},{"key":"2292_CR9","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511607219","volume-title":"Central simple algebras and Galois cohomology","author":"P Gille","year":"2006","unstructured":"Gille, P., Szamuely, T.: Central simple algebras and Galois cohomology, vol. 101. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2006)"},{"issue":"1","key":"2292_CR10","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1090\/S0002-9939-08-09450-1","volume":"137","author":"A Hogadi","year":"2009","unstructured":"Hogadi, A.: Products of Brauer-Severi surfaces. Proc. AMS 137(1), 45\u201350 (2009)","journal-title":"Proc. AMS"},{"key":"2292_CR11","doi-asserted-by":"crossref","unstructured":"Keller, B.: On differential graded categories. International congress of mathematicians (Madrid), Vol.\u00a0II, pp. 151\u2013190. The European Mathematical Society, Z\u00fcrich (2006)","DOI":"10.4171\/022-2\/8"},{"key":"2292_CR12","series-title":"With a preface in French by J. Tits","volume-title":"The book of involutions","author":"M-A Knus","year":"1998","unstructured":"Knus, M.-A., Merkurjev, A., Rost, M., Tignol, J.-P.: The book of involutions. With a preface in French by J. Tits, vol. 44. AMS Colloquium Publications, Providence (1998)"},{"issue":"1","key":"2292_CR13","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1016\/j.aim.2005.01.004","volume":"198","author":"J Koll\u00e1r","year":"2005","unstructured":"Koll\u00e1r, J.: Conics in the Grothendieck ring. Adv. Math. 198(1), 27\u201335 (2005)","journal-title":"Adv. Math."},{"key":"2292_CR14","series-title":"Cambridge Studies in Advanced Mathematics","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511734991","volume-title":"Rational and nearly rational varieties","author":"J Koll\u00e1r","year":"2004","unstructured":"Koll\u00e1r, J., Smith, K., Corti, A.: Rational and nearly rational varieties. Cambridge Studies in Advanced Mathematics, vol. 92. Cambridge University Press, Cambridge (2004)"},{"key":"2292_CR15","unstructured":"Kontsevich, M.: Noncommutative motives. Talk at the IAS on the occasion of the $$61^{\\rm st}$$ birthday of Pierre Deligne. Available at http:\/\/video.ias.edu\/Geometry-and-Arithmetic (2005)"},{"key":"2292_CR16","doi-asserted-by":"crossref","unstructured":"Kontsevich, M.: Notes on motives in finite characteristic. Algebra, arithmetic, and geometry: in honor of Yu I Manin. Vol. II, pp. 213\u2013247, Progr. Math., 270, Birkh\u00e4user, Boston (2009)","DOI":"10.1007\/978-0-8176-4747-6_7"},{"key":"2292_CR17","unstructured":"Kontsevich, M.: Mixed noncommutative motives. Talk at the workshop on homological mirror symmetry, Miami. Available at http:\/\/www-math.mit.edu\/auroux\/frg\/miami10-notes (2010)"},{"key":"2292_CR18","series-title":"Graduate Studies in Mathematics","volume-title":"Introduction to quadratic forms over fields","author":"T-Y Lam","year":"2005","unstructured":"Lam, T.-Y.: Introduction to quadratic forms over fields. Graduate Studies in Mathematics, vol. 67. American Mathematical Society, Providence (2005)"},{"issue":"1","key":"2292_CR19","doi-asserted-by":"publisher","first-page":"85","DOI":"10.17323\/1609-4514-2003-3-1-85-95","volume":"3","author":"M Larsen","year":"2003","unstructured":"Larsen, M., Lunts, V.: Motivic measures and stable birational geometry. Mosc. Math. J. 3(1), 85\u201395 (2003)","journal-title":"Mosc. Math. J."},{"key":"2292_CR20","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/s002290050199","volume":"100","author":"D Lewis","year":"1999","unstructured":"Lewis, D., Tignol, J.-P.: Classification theorems for central simple algebras with involution (with an appendix by R. Parimala). Manuscr. Math. 100, 259\u2013276 (1999)","journal-title":"Manuscr. Math."},{"issue":"4","key":"2292_CR21","first-page":"852","volume":"56","author":"D Orlov","year":"1992","unstructured":"Orlov, D.: Projective bundles, monoidal transformations, and derived categories of coherent sheaves. Izv. Ross. Akad. Nauk. Ser. Mat. 56(4), 852\u2013862 (1992)","journal-title":"Izv. Ross. Akad. Nauk. Ser. Mat."},{"issue":"6","key":"2292_CR22","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1007\/BF00961020","volume":"8","author":"I Panin","year":"1994","unstructured":"Panin, I.: On the algebraic $$K$$-theory of twisted flag varieties. K-Theory 8(6), 541\u2013585 (1994)","journal-title":"K-Theory"},{"issue":"215","key":"2292_CR23","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1515\/crll.1964.214-215.207","volume":"214","author":"P Roquette","year":"1964","unstructured":"Roquette, P.: Isomorphisms of generic splitting fields of simple algebras. J. Reine Angew. Math. 214(215), 207\u2013226 (1964)","journal-title":"J. Reine Angew. Math."},{"key":"2292_CR24","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1016\/0040-9383(75)90025-7","volume":"14","author":"R Steinberg","year":"1975","unstructured":"Steinberg, R.: On a theorem of Pittie. Topology 14, 173\u2013177 (1975)","journal-title":"Topology"},{"key":"2292_CR25","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1016\/j.jalgebra.2014.06.028","volume":"417","author":"G Tabuada","year":"2014","unstructured":"Tabuada, G.: Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives. J. Algebr. 417, 15\u201338 (2014)","journal-title":"J. Algebr."},{"key":"2292_CR26","series-title":"University Lecture Series.","doi-asserted-by":"crossref","DOI":"10.1090\/ulect\/063","volume-title":"Noncommutative Motives. With a preface by Yuri I. Manin","author":"G Tabuada","year":"2015","unstructured":"Tabuada, G.: Noncommutative Motives. With a preface by Yuri I. Manin. University Lecture Series., vol. 63. American Mathematical Society, Providence (2015)"},{"key":"2292_CR27","doi-asserted-by":"publisher","first-page":"648","DOI":"10.1016\/j.aim.2019.04.020","volume":"349","author":"G Tabuada","year":"2019","unstructured":"Tabuada, G.: Noncommutative motives in positive characteristic and their applications. Adv. Math. 349, 648\u2013681 (2019)","journal-title":"Adv. Math."},{"key":"2292_CR28","doi-asserted-by":"publisher","first-page":"1122","DOI":"10.1016\/j.aim.2016.08.031","volume":"303","author":"G Tabuada","year":"2016","unstructured":"Tabuada, G., Van den Bergh, M.: Noncommutative motives of separable algebras. Adv. Math. 303, 1122\u20131161 (2016)","journal-title":"Adv. Math."},{"issue":"1","key":"2292_CR29","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1090\/tran\/6956","volume":"370","author":"G Tabuada","year":"2018","unstructured":"Tabuada, G., Van den Bergh, M.: The Gysin triangle via localization and $${\\mathbb{A}}^1$$-homotopy invariance. Trans. AMS 370(1), 421\u2013446 (2018)","journal-title":"Trans. AMS"},{"key":"2292_CR30","first-page":"196","volume":"247","author":"J Tits","year":"1971","unstructured":"Tits, J.: Repr\u00e9sentations lin\u00e9aires irr\u00e9ductibles d\u2019un groupe r\u00e9ductif sur un corps quelconque. J. Reine Angew. Math. 247, 196\u2013220 (1971)","journal-title":"J. Reine Angew. Math."},{"key":"#cr-split#-2292_CR31.1","doi-asserted-by":"crossref","unstructured":"Tregub, S.: Birational equivalence of Brauer-Severi manifolds. Uspekhi Mat. Nauk 46(6), 217-218 (1991)","DOI":"10.1070\/RM1991v046n06ABEH002866"},{"key":"#cr-split#-2292_CR31.2","doi-asserted-by":"crossref","unstructured":"Transl. Russ. Math. Surv. 46 (6), 229 (1991)","DOI":"10.1070\/RM1991v046n06ABEH002866"}],"container-title":["Mathematische Annalen"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00208-021-02292-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00208-021-02292-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00208-021-02292-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,14]],"date-time":"2023-01-14T15:52:16Z","timestamp":1673711536000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00208-021-02292-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,6]]},"references-count":32,"journal-issue":{"issue":"3-4","published-print":{"date-parts":[[2022,4]]}},"alternative-id":["2292"],"URL":"https:\/\/doi.org\/10.1007\/s00208-021-02292-6","relation":{},"ISSN":["0025-5831","1432-1807"],"issn-type":[{"type":"print","value":"0025-5831"},{"type":"electronic","value":"1432-1807"}],"subject":[],"published":{"date-parts":[[2021,11,6]]},"assertion":[{"value":"18 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 September 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 October 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 November 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author states that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}