On Crossed Product Algebras over Henselian Valued Fields

Let D be a tame central division algebra over a Henselian valued field E, [Formula: see text] be the residue division algebra of D, [Formula: see text] be the residue field of E, and n be a positive integer. We prove that Mn([Formula: see text]) has a strictly maximal subfield which is Galois (resp., abelian) over [Formula: see text] if and only if Mn(D) has a strictly maximal subfield K which is Galois (resp., abelian) and tame over E with ΓK ⊆ ΓD, where ΓK and ΓD are the value groups of K and D, respectively. This partially generalizes the result proved by Hanke et al. in 2016 for the case n = 1.

Radicability condition on GLn(D)

Given an indivisible field [Formula: see text], let [Formula: see text] be a finite dimensional noncommutative [Formula: see text]-central division algebra. It is shown that if [Formula: see text] is radicable, then [Formula: see text] is the ordinary quaternion division algebra and [Formula: see text] is divisible. Also, it is shown that when [Formula: see text] is a field of characteristic zero and [Formula: see text], then [Formula: see text] is radicable if and only if for any field extension [Formula: see text] with [Formula: see text], [Formula: see text] is divisible.

Existence of Nonabelian Free Subgroups in the Maximal Subgroups of GLn(D)

Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of GLn(D) contains a non-cyclic free subgroup. In general, it is shown that either M contains a non-cyclic free subgroup or there exists a unique maximal subfield K of Mn(D) such that NGLn(D)(K*)=M, K* ◁ M, K/F is Galois with Gal (K/F) ≅ M/K*, and F[M]=Mn(D). In particular, when F is global or local, it is proved that if ([D:F], Char (F))=1, then every non-abelian maximal subgroup of GL1(D) contains a non-cyclic free subgroup. Furthermore, it is also shown that GLn(F) contains no solvable maximal subgroups provided that F is local or global and n ≥ 5.

On some arithmetic properties of automorphic forms ofGLmover a division algebra

In this paper we investigate arithmetic properties of automorphic forms on the group G' = GLm/D, for a central division-algebra D over an arbitrary number field F. The results of this article are generalizations of results in the split case, i.e. D = F, by Shimura, Harder, Waldspurger and Clozel for square-integrable automorphic forms and also by Franke and Franke–Schwermer for general automorphic representations. We also compare our theorems on automorphic forms of the group G′ to statements on automorphic forms of its split form using the global Jacquet–Langlands correspondence developed by Badulescu and Badulescu–Renard. Beside that we prove that the local version of the Jacquet–Langlands transfer at an archimedean place preserves the property of being cohomological.

SLn, Orthogonality Relations and Transfer

Let π be a square integrable representation of G′ = SLn(D), with D a central division algebra of finite dimension over a local field F of non-zero characteristic. We prove that, on the elliptic set, the character of π equals the complex conjugate of the orbital integral of one of the pseudocoefficients of π. We prove also the orthogonality relations for characters of square integrable representations of G′. We prove the stable transfer of orbital integrals between SLn(F) and its inner forms.

Nicely semiramified division algebras over Henselian fields

This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite-dimensional central division algebra over a Henselian fieldEwith an inertial maximal subfield and a totally ramified maximal subfield (not necessarily of radical type) (resp., split by inertial and totally ramified field extensions ofE) is nicely semiramified.

A Result on Derivations with Algebraic Values

Let R be a prime algebra over a field F and let d be a non-zero derivation in R such that for every x ∊ R, d(x) is algebraic over F of bounded degree. Then R is a primitive ring with a minimal right ideal eR, where e2 = e and eRe is a finite dimensional central division algebra.

Construction of arithmetic automorphic functions for special Clifford groups

An important problem in the theory of arithmetic automorphic functions is to construct, for a reductive algebraic group over Q which defines a bounded symmetric domain, a system of canonical models [2], [6], [18]. For the similitude group of a hermitian form over a quaternion algebra whose center is a totally real field, this is solved by Shimura [17], and for the similitude group of a hermitian form with respect to an involution of the second kind of a central division algebra over a CM-field, by Miyake [8], In this paper, we show that this also can be done for the special Clifford group of a quadratic form Q over a totally real algebraic number field. (We have to impose certain conditions on the signature of Q in order that G defines a bounded symmetric domain, see 1.1.)