Jacques Tits motivic measure release_3el5hftjkvd7vi5npaezymprbi

Abstract

<jats:title>Abstract</jats:title>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.
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