Symplectic involutions, quadratic pairs and function fields of conics
نویسندگان
چکیده
منابع مشابه
Discriminant of Symplectic Involutions
We define an invariant of torsors under adjoint linear algebraic groups of type Cn—equivalently, central simple algebras of degree 2n with symplectic involution—for n divisible by 4 that takes values in H(k, μ2). The invariant is distinct from the few known examples of cohomological invariants of torsors under adjoint groups. We also prove that the invariant detects whether a central simple alg...
متن کاملPairs of Quadratic Forms over Finite Fields
Let Fq be a finite field with q elements and let X be a set of matrices over Fq. The main results of this paper are explicit expressions for the number of pairs (A,B) of matrices in X such that A has rank r, B has rank s, and A + B has rank k in the cases that (i) X is the set of alternating matrices over Fq and (ii) X is the set of symmetric matrices over Fq for odd q. Our motivation to study ...
متن کاملSums of Squares in Function Fields of Quadrics and Conics
For a quadric Q over a real field k, we investigate whether finiteness of the Pythagoras number of the function field k(Q) implies the existence of a uniform bound on the Pythagoras numbers of all finite extensions of k. We give a positive answer if the quadratic form that defines Q is weakly isotropic. In the case where Q is a conic, we show that the Pythagoras number of k(Q) is 2 only if k is...
متن کاملConics over function fields and the Artin-Tate conjecture
We prove that the Hasse principle for conics over function fields is a simple consequence of a provable case of the Artin-Tate conjecture for surfaces over finite fields. Hasse proved that a conic over a global field has a rational point if and only if it has points over all completions of the global field, an instance of the so-called local-global or Hasse principle. The case of the rational n...
متن کاملDecomposable Quadratic Forms and Involutions
In his book on compositions of quadratic forms, Shapiro asks whether a quadratic form decomposes as a tensor product of quadratic forms when its adjoint involution decomposes as a tensor product of involutions on central simple algebras. We give a positive answer for quadratic forms defined over local or global fields and produce counterexamples over fields of rational fractions in two variable...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Pure and Applied Algebra
سال: 2017
ISSN: 0022-4049
DOI: 10.1016/j.jpaa.2016.10.016