The universal affine extension of an abelian variety

Every abelian variety over a field k admits a universal
extension by an affine k-group scheme. The talk will present
a construction of this universal affine extension, first due to
Serre when k is algebraically closed of characteristic zero.
We will then discuss its structure, with applications to the
category of commutative quasi-compact group schemes, and to
homogeneous vector bundles over abelian varieties.