Additional structure on algebraic groups in Hasse fields

Model theory, a branch of mathematical logic which involves the study of sets definable in a mathematical structure using an appropriate first order language, has recently shown its capacity to produce applications to other domains of mathematics, principally to algebra, algebraic geometry and number theory. The main example is the proof by Ehud Hrushovski of a conjecture of algebraic geometry, known as the Mordell-Lang conjecture for function fields (see [Hru96]). One point of this proof was to augment the usual language of fields, suitable for algebraic geometry, via new operators, essentially by adding a Hasse derivation. Our aim here is to describe, in a geometric fashion, the new objects which are definable after adding a Hasse derivation. We develop the relevant context of Dalgebraic geometry, and we concentrate on the description of “small” (rationally thin) D-algebraic subgroups of algebraic groups (i.e. objects of “finite dimension” in a strong sense, in a universe where each point comes with the infinite sequence of its derivatives). As Alexandru Buium did for the characteristic zero case in [Bui92], we relate these small subgroups to the notion of D-structure on an algebraic group (Theorem 3). Then we obtain some consequences about the existence of D-algebraic subgroups which are both rationally thin and Zariski-dense in a given algebraic group. By a precise study of the conditions under which an abelian variety admits a D-structure, we obtain, in the positive characteristic case, a link between the field of definition of an abelian variety and rational thinness of the subgroups of its divisible points in a given Hasse field (Corollary 4). This first account gives only basic ideas of the objects described, and sketches of the proofs. Further publications on this topic are in preparation, and will provide additional details concerning D-algebraic geometry, including a schematic treatment, and its applications to the study of algebraic groups in positive characteristic.