De Beaux Groupes

TL;DR

This paper characterizes interpretable groups in algebraically closed fields with a distinguished subfield, showing they are essentially extensions of algebraic groups' rational points by interpretable quotients.

Contribution

It provides a detailed description of interpretable groups in beautiful pairs of algebraically closed fields, extending previous understanding of their structure.

Findings

Interpretable groups are, up to isogeny, extensions of E-rational points of algebraic groups.

Any interpretable group is a quotient of an algebraic group by E-rational points of a subgroup.

The structure theorem applies to beautiful pairs of algebraically closed fields.

Abstract

In this short paper, we will provide a characterisation of interpretable groups in a beautiful pair (K, E) of algebraically closed fields : every interpretable group is, up to isogeny, the extension of the subgroup of E-rational points of an algebraic group by an interpretable group which is the quotient of an algebraic group by the E-rational points of an algebraic subgroup.---Dans une belle paire (K;E) de corps alg\'ebriquement clos, un groupe d\'efinissable se projette, \`a isog\'enie pr\`es, sur les points E-rationnels d'un groupe alg\'ebrique ayant pour noyau un groupe alg\'ebrique. Un groupe interpr\'etable est, \`a isog\'enie pr\`es, l'extension des points E-rationnels d'un groupe alg\'ebrique par un groupe interpr\'etable, qui est lui le quotient d'un groupe alg\'ebrique par les points E-rationnels d'un sous-groupe alg\'ebrique.