On abstract homomorphisms of algebraic groups

TL;DR

This paper investigates the structure of abstract homomorphisms between rational points of algebraic groups, especially over various fields, extending known results to anisotropic and non-reductive cases.

Contribution

It provides new insights into homomorphisms from algebraic groups over specific fields, including anisotropic groups over non-archimedean local fields, and generalizes existing structural results.

Findings

Results on homomorphisms from algebraic groups over number fields

Analysis of homomorphisms from anisotropic groups over local fields

Structural theorems for homomorphisms involving unipotent and solvable groups

Abstract

In this paper we study abstract group homomorphisms between the groups of rational points of linear algebraic groups which are not necessarily reductive. One of our main goal is to obtain results on homomorphisms from the groups of rational points of linear algebraic groups defined over certain specific fields to the groups of rational points of linear algebraic groups over number fields and non-archimedean local fields of characteristic zero; in this set-up we deal with the unexplored topic of abstract homomorphisms from the groups of rational points of anisotropic groups over non-archimedean local fields of characteristic zero. We also obtain results on abstract homomorphisms from unipotent and solvable groups, and prove general results on the structures of abstract homomorphisms using the celebrated result of Borel and Tits in this area and a well-known theorem due to Tits on the structure of the groups of rational points of isotropic semisimple groups.