Criteria of motivic equivalence for quadratic forms and central simple algebras

AbstractIn this article we construct a new motivic measure called the Jacques Tits motivic measure. 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 $$\{3, 4, 5, 6\}$$ { 3 , 4 , 5 , 6 } , 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 k with $$I^3(k)=0$$ I 3 ( k ) = 0 , 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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Applications of patching to quadratic forms and central simple algebras

Inventiones mathematicae ◽

10.1007/s00222-009-0195-5 ◽

2009 ◽

Vol 178 (2) ◽

pp. 231-263 ◽

Cited By ~ 32

Author(s):

David Harbater ◽

Julia Hartmann ◽

Daniel Krashen

Keyword(s):

Quadratic Forms ◽

Central Simple Algebras ◽

Simple Algebras ◽

Central Simple

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An identification system based on the explicit isomorphism problem

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-021-00529-0 ◽

2021 ◽

Author(s):

Sándor Z. Kiss ◽

Péter Kutas

Keyword(s):

Quadratic Forms ◽

Isomorphism Problem ◽

Identification System ◽

Division Algebras ◽

Central Simple Algebras ◽

Zero Knowledge ◽

Algorithmic Problems ◽

Simple Algebras ◽

Central Simple ◽

Integral Equivalence

AbstractWe propose a new identification system based on algorithmic problems related to computing isomorphisms between central simple algebras. We design a statistical zero knowledge protocol which relies on the hardness of computing isomorphisms between orders in division algebras which generalizes a protocol by Hartung and Schnorr, which relies on the hardness of integral equivalence of quadratic forms.

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Hermitian forms over odd dimensional algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500006994 ◽

1989 ◽

Vol 32 (1) ◽

pp. 139-145

Author(s):

D. W. Lewis

Keyword(s):

Galois Group ◽

Quadratic Forms ◽

Natural Extension ◽

Central Simple Algebras ◽

Witt Group ◽

Hermitian Forms ◽

Odd Degree ◽

Simple Algebras ◽

Central Simple ◽

Suitable Restriction

Let L be an odd degree extension field of the field K, char K ≠ 2. Let U* denote the natural extension map from W(K) to W(L) where W(K), resp. W(L) denotes the Witt group of quadratic forms over K, resp. L. It is well-known that U* is injective [4, p. 198]. In fact Springer [10] proved a stronger theorem, namely that if φ is anisotropic over K then it remains anisotropic on extension to L. Rosenberg and Ware [8] proved that if L is a Galois extension then the image of U* is precisely the subgroup of W(L) fixed by the Galois group of L over K, this Galois group having a natural action on W(L). See [4, p. 214] and [3] for a quick proof. See also Dress [1] who extended these results to equivariant forms. In this article we investigate the corresponding map U* when we replace the field L by a central simple K-algebra of odd degree and indeed more generally by any finite dimensional K-algebra which becomes odd-dimensional on factoring out by the radical. Our algebras are equipped with an involution of the second kind, i.e. one which is non-trivial on the centre, and we replace quadratic forms by hermitian forms with respect to the involution. We show that U* is injective for all the algebras mentioned above and that a weaker version of Springer's theorem holds for central simple algebras of odd degree provided we make a suitable restriction on the nature of the involution. We show that the analogue of the Rosenberg-Ware result is valid for hermitian forms over odd-dimensional Galois field extensions but that for central simple algebras of odd degree a result as nice as the Rosenberg–Ware one cannot hold. Indeed the group of all K-automorphisms of such an algebra which commute with the involution fixes all of the Witt group. However the map U* is not surjective in general even for division algebras of odd degree.

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Patching over Berkovich curves and quadratic forms

Compositio Mathematica ◽

10.1112/s0010437x19007632 ◽

2019 ◽

Vol 155 (12) ◽

pp. 2399-2438

Author(s):

Vlerë Mehmeti

Keyword(s):

Quadratic Forms ◽

Sufficient Conditions ◽

Analytic Geometry ◽

Central Simple Algebras ◽

Local Isotropy ◽

Simple Algebras ◽

Central Simple ◽

Global Principle ◽

Analytic Curves ◽

Invent Math

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.

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