Linear algebraic groups are groups of matrices that are also algebraic varieties, meaning they are defined by polynomial equations. These structures play a crucial role in connecting algebraic geometry with group theory, providing insights into symmetries and transformations in various mathematical contexts.
Weil divisors are formal sums of codimension-one subvarieties on an algebraic variety, playing a crucial role in the study of algebraic geometry by capturing information about the variety's structure. They help in defining line bundles and are essential in the formulation of the divisor class group, which is central to understanding the variety's geometric and arithmetic properties.
An Abelian variety is a complete algebraic variety that is also a group, meaning it has a group structure compatible with its geometric structure. These varieties play a central role in number theory and algebraic geometry, particularly in the study of elliptic curves, modular forms, and the formulation of the Langlands program.