On some local–global principles for linear algebraic groups over infinite algebraic extensions of global fields

Abstract

In this paper, we develop an arithmetic theory of quadratic and hermitian forms over infinite algebraic extensions of local and global fields. In particular, we prove that the cohomological Hasse principle for H1 holds for all semisimple simply connected algebraic groups defined over any infinite algebraic extension of any global field and we also show the validity of some local–global principles for (skew-)hermitian forms defined over such infinite extension fields. As applications, we deduce some analogs of well known results such as Landherr's and Kneser's Strong Hasse principle, Hasse–Maass–Schilling Norm Theorem, Albert–Brauer–Hasse–Noether Theorem and Hasse Norm Theorem over such fields.