2010/08/17 by Maneesh Thakur, Thakur, Maneesh
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #FOS: Mathematics #Group Theory (math.GR) #Primary 20G15 #Secondary 17C30
paper · pdf · doi:10.48550/arxiv.1008.2919
openalex publication_date 2010/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be an arbitrary field. The main aim of this paper is to prove the Tits-Weiss conjecture for Albert division algebras over k which are pure first Tits constructions. This conjecture asserts that for an Albert division algebra A over a field k, every norm similarity of A is inner modulo scalar multiplications. It is known that k-forms of E8 with index E788,2 and anisotropic kernel a strict inner k-form of E6 correspond bijectively (via Moufang hexagons) to Albert division algebras over k. The Kneser-Tits problem for a form of E8 as above is equivalent to the Tits-Weiss conjecture (see \citeTW). Hence we provide a solution to the Kneser-Tits problem for forms of E8 arising from pure first Tits construction Albert division algebras. As an application, we prove that for G=\bf Aut(A),~G(k)/R=1, where A is a pure first construction Albert division algebra over k and R stands for R-equivalence in the sense of Manin (\citeM).