Embedding functors and their arithmetic properties

TL;DR

This paper studies the embedding of tori into reductive groups via embedding functors, establishing their representability, analyzing their properties over local fields, and applying these results to classical algebraic problems.

Contribution

It introduces the concept of embedding functors and orientations, proves their representability and homogeneity, and applies these to the local-global principle and classical algebraic theorems.

Findings

Embedding functors are representable and homogeneous spaces.

Orientation and Tits index determine embedding existence over local fields.

The Brauer-Manin obstruction is the only obstruction to the local-global principle.

Abstract

In this article, we focus on how to embed a torus $\rT$ into a reductive group $\rG$ with respect to a given root datum $\Psi$ over a scheme $\rS$. This problem also relates to how to embed an \'etale algebra with involution into a central simple algebra with involution (cf. \cite{PR1}). We approach this problem by defining the embedding functor, and prove that the embedding functor is representable and is a left homogeneous space over $\rS$ under the automorphism group of $\rG$. In order to fix a connected component of the embedding functor, we define an orientation $u$ of $\Psi$ with respect to $\rG$. We show that the oriented embedding functor is also representable and is a homogeneous space under the adjoint action of $\rG$. Over a local field, the orientation $u$ and the Tits index of $\rG$ determine the existence of embedding of $\rT$ into $\rG$ with respect to the given root datum $\Psi$. We also use the techniques developed in Borovoi's paper \cite{Bo} to prove that the local-global principle holds for oriented embedding functors in certain cases. Actually, the Brauer-Manin obstruction is the only obstruction to the local-global principle for the oriented embedding functor. Finally, we apply the results of oriented embedding functors to give an alternative proof of Prasad and Rapinchuk's Theorem, and improve Theorem 7.3 in \cite{PR1}.