De Beaux Groupes
Thomas Blossier (ICJ), Amador Martin-Pizarro (ICJ)

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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Advanced Topology and Set Theory
