Pseudo-abelian varieties

TL;DR

This paper introduces pseudo-abelian varieties as a new class of algebraic groups over arbitrary fields, extending classical results and exploring their structure, properties, and connections to other group types.

Contribution

It defines pseudo-abelian varieties, analyzes their structure, and relates them to unipotent and pseudo-reductive groups, extending classical theorems to non-perfect fields.

Findings

Pseudo-abelian varieties generalize abelian varieties over arbitrary fields.

Many properties of abelian varieties, like Mordell-Weil, extend to pseudo-abelian varieties.

Conjecture on describing Ext^2(G_a,G_m) over any field.

Abstract

Chevalley's theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian variety over an arbitrary field k to be a smooth connected k-group in which every smooth connected affine normal k-subgroup is trivial. This gives a new point of view on the classification of algebraic groups: every smooth connected group over a field is an extension of a pseudo-abelian variety by a smooth connected affine group, in a unique way. We work out much of the structure of pseudo-abelian varieties. These groups are closely related to unipotent groups in characteristic p and to pseudo-reductive groups as studied by Tits and Conrad-Gabber-Prasad. Many properties of abelian varieties such as the Mordell-Weil theorem extend to pseudo-abelian varieties. Finally, we conjecture a description of Ext^2(G_a,G_m) over any field by generators and relations, in the spirit of the Milnor conjecture.