Editing Linear algebraic group (section)

===Abelian varieties===
[[Algebraic group]]s which are not affine behave very differently. In particular, a smooth connected group scheme which is a projective variety over a field is called an [[abelian variety]]. In contrast to linear algebraic groups, every abelian variety is commutative. Nonetheless, abelian varieties have a rich theory. Even the case of [[elliptic curve]]s (abelian varieties of dimension 1) is central to [[number theory]], with applications including the proof of [[Fermat's Last Theorem]].