The Norm of a Skew Polynomial

Let D be a finite-dimensional division algebra over its center and R = D[t;σ,δ] a skew polynomial ring. Under certain assumptions on δ and σ, the ring of central quotients D(t;σ,δ) = {f/g|f ∈ D[t;σ,δ],g ∈ C(D[t;σ,δ])} of D[t;σ,δ] is a central simple algebra with reduced norm N. We calculate the norm N(f) for some skew polynomials f ∈ R and investigate when and how the reducibility of N(f) reflects the reducibility of f.

A generalisation of Amitsur’s A-polynomials

We find examples of polynomials f ∈ D [t; σ, δ] whose eigenring ℰ(f) is a central simple algebra over the field F = C ∩ Fix(σ) ∩ Const(δ).

Splitting quaternion algebras defined over a finite field extension

We study systems of quadratic forms over fields and their isotropy over [Formula: see text]-extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a finite field extension. As a consequence we obtain that every central simple algebra of degree [Formula: see text] is split by a [Formula: see text]-extension of degree at most [Formula: see text].

Multiplicity One for Pairs of Prasad–Takloo-Bighash Type

Let ${\textrm{E}}/{\textrm{F}}$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let ${\textrm{A}}$ be an ${\textrm{F}}$-central simple algebra of even dimension so that it contains ${\textrm{E}}$ as a subfield, set ${\textrm{G}}={\textrm{A}}^\times $ and ${\textrm{H}}$ for the centralizer of ${\textrm{E}}^\times $ in ${\textrm{G}}$. Using a Galois descent argument, we prove that all double cosets ${\textrm{H}} g {\textrm{H}}\subset{\textrm{G}}$ are stable under the anti-involution $g\mapsto g^{-1}$, reducing to Guo’s result for ${\textrm{F}}$-split ${\textrm{G}}$ [14], which we extend to fields of positive characteristic different from $2$. We then show, combining global and local results, that ${\textrm{H}}$-distinguished irreducible representations of ${\textrm{G}}$ are self-dual and this implies that $({\textrm{G}},{\textrm{H}})$ is a Gelfand pair $$\begin{equation*}\operatorname{dim}_{\mathbb{C}}(\operatorname{Hom}_{{\textrm{H}}}(\pi,\mathbb{C}))\leq 1\end{equation*}$$for all smooth irreducible representations $\pi $ of ${\textrm{G}}$. Finally we explain how to obtain the multiplicity one statement in the archimedean case using the criteria of Aizenbud and Gourevitch ([1]), and we then show self-duality of irreducible distinguished representations in the archimedean case too.

SOLVABLE CROSSED PRODUCT ALGEBRAS REVISITED

For any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF(M), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit and overlaps with a similar result by Albert which, however, was not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F.

Motivic decomposition of compactifications of certain group varieties

Let D be a central simple algebra of prime degree over a field and let E be an {\operatorname{\mathbf{SL}}_{1}(D)} -torsor. We determine the complete motivic decomposition of certain compactifications of E. We also compute the Chow ring of E.

Isotropy of orthogonal involutions in characteristic two

A totally singular quadratic form is associated to any central simple algebra with orthogonal involution in characteristic two. It is shown that the given involution is isotropic if and only if its corresponding quadratic form is isotropic.

The Coniveau Filtration on for Some Severi–Brauer Varieties

We produce an isomorphism $E_{\infty }^{m,-m-1}\cong \text{Nrd}_{1}(A^{\otimes m})$ between terms of the $\text{K}$-theory coniveau spectral sequence of a Severi–Brauer variety $X$ associated with a central simple algebra $A$ and a reduced norm group, assuming $A$ has equal index and exponent over all finite extensions of its center and that $\text{SK}_{1}(A^{\otimes i})=1$ for all $i>0$.