On abstract homomorphisms of algebraic groups

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.