Weak isotropy of central simple algebras with orthogonal involutions over totally positive field extensions
Priyabrata Mandal, Abhay Soman
TL;DR
This work probes how totally positive field extensions influence the weak isotropy of orthogonal involutions on central simple algebras. It establishes that isotropy over a totally positive extension does not universally descend to weak isotropy over the base field, via a carefully constructed counterexample, while proving a descent result for finite Galois $2$-extensions. It also confirms Becher's conjecture in a concrete setting: when the base is extended by a Galois totally positive field and the division algebra has index $2^n$ and exponent $2$ containing a suitable subfield of $F_{py}$, the Pythagorean index remains invariant under tensoring with the extension. Collectively, the results illuminate the nuanced interaction among orderings, tp extensions, and involutions in the landscape of real fields and division algebras, advancing the theory of involutions over formally real settings.
Abstract
In this paper, we explore the behavior of orthogonal involutions in the context of totally positive field extensions. Let $K/F$ be a totally positive extension of formally real fields. By Becher's result, if a quadratic form $q$ over $F$ becomes isotropic over $K$, then $q$ is weakly isotropic over $F$. We present an example in which, despite $K/F$ being totally positive, a central simple algebra $(A,σ)$ over $F$ with an orthogonal involution becomes isotropic over $K$, while remaining strongly isotropic over $F$. However, when $K/F$ is assumed to be a Galois totally positive $2$-extension of formally real fields, we show that an analogue of Becher's result for quadratic forms holds for orthogonal involutions. Furthermore, for a totally positive Galois field extension $K/F$, we verify Becher's conjecture for central division algebras of index $2^n$ and exponent $2$ containing a subfield of $F_{py}$ of degree $2^{n-2}$ over $F$.