Applications of patching to quadratic forms and central simple algebras

This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over Henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh (in Preprint , 2007) on the u-invariant of p-adic function fields, p≠2. The strategy relies on a local-global principle for homogeneous spaces for rational algebraic groups, combined with local computations.     

Witt equivalence of central simple algebras with involution

The notion of Witt equivalence of central simple algebras with involution is introduced. It is shown that the standard invariants, i.e. the discriminant, the signature and the Clifford algebra, depend only on the Witt class of the algebra with involution. For a given filedF the tensor product is used to construct a semigroup\(\tilde S\left( F \right)\) and this semigroup is shown to have properties analogous to the multiplicative properties of the Witt ring of quadratic forms overF.     

Witt equivalence of central simple algebras with involution

The notion of Witt equivalence of central simple algebras with involution is introduced. It is shown that the standard invariants, i.e. the discriminant, the signature and the Clifford algebra, depend only on the Witt class of the algebra with involution. For a given filedF the tensor product is used to construct a semigroup and this semigroup is shown to have properties analogous to the multiplicative properties of the Witt ring of quadratic forms overF.     

Spinor equivalence of quadratic forms

Let f be an integral quadratic form in three or more variables and g any form in the genus of f. There exist an effectively determinable prime p and a form g′, belonging to the proper spinor genus of g, such that g′ is a p-neighbor of f in the graph of f. Using this, an alternative decision procedure for the spinor equivalence of quadratic forms is given.     

Higher trace forms and essential dimension in central simple algebras

We show that the essential dimension of a finite-dimensional central simple algebra coincides with the essential dimension of its r-linear trace form, for any r ≥ 3. Received: 15 March 2006     

Tame algebras and Tits quadratic forms

We solve a long standing problem concerning the connection between the tameness of simply connected algebras and the weak nonnegativity of the associated Tits integral quadratic forms, and derive some consequences.     

The equivalence problem and canonical forms for quadratic Lagrangians

The general equivalence and canonical form problems for quadratic variational problems under arbitrary linear changes of variable are formulated, and the role of classical invariant theory in their general solution is made clear. A complete solution to both problems for planar, first order quadratic variational problems is provided, including a complete list of canonical forms for the Lagrangians and corresponding Euler-Lagrange equations. Algorithmic procedures for determining the equivalence class and the explicit canonical form of a given Lagrangian are provided. Applications to planar anisotropic elasticity are indicated.     

Motivic equivalence of quadratic forms. II

Let F be a field of characteristic ≠2 and φ be a quadratic form over F. By X _φ we denote the projective variety given by the equation φ=0. For each positive even integer d≥8 (except for d=12) we construct a field F and a pair φ, ψ of anisotropic d-dimensional forms over F such that the Chow motives of X _φ and X _ψ coincide but . For a pair of anisotropic (2 ^{n -1})-dimensional quadrics X and Y, we prove that existence of a rational morphism Y→X is equivalent to existence of a rational morphism Y→X. Received: 27 September 1999 / Revised version: 27 December 1999     

Transfer of quadratic forms and of quaternion algebras over quadratic field extensions

Two different proofs are given showing that a quaternion algebra Q defined over a quadratic étale extension K of a given field has a corestriction that is not a division algebra if and only if Q contains a quadratic algebra that is linearly disjoint from K. This is known in the case of a quadratic field extension in characteristic different from two. In the case where K is split, the statement recovers a well-known result on biquaternion algebras due to Albert and Draxl.     

Freeness of linear and quadratic forms in von Neumann algebras

We characterize the semicircular distribution by freeness of linear and quadratic forms in noncommutative random variables from tracial W^?-probability spaces with relaxed moment conditions.     

Essential dimension of central simple algebras

Let p be a prime integer, 1≤s≤r be integers and F be a field of characteristic different from p. We find upper and lower bounds for the essential p-dimension ed _p (\( Al{{g}_{{{{p}^r},{{p}^s}}}} \)) of the class \( Al{{g}_{{{{p}^r},{{p}^s}}}} \) of central simple algebras of degree p ^r and exponent dividing p ^s . In particular, we show that ed(Alg _8,2)=ed₂(Alg _8,2)=8 and ed _p (\( Al{{g}_{{{{p}^2},p}}} \))=p ²+p for p odd.