An abstract Torelli theorem

We consider the abstract pair 'group J with a distinguished subset C' where J is the K-points of a Jacobian of a smooth curve C, K algebraically closed. We prove that this pair determines the field, the curve and the Jacobian uniquely up to an isomorphism of fields and a bijective morphism of the varieties, which is a bijective isogeny on Jacobians. In characteristic 0 the bijective morphism is an isomorphism. Over finite fields the theorem proves a conjecture from the recent paper by Bogomolov, Korotaev and Tschinkel. The proof is model-theoretic and is based on a theorem of Rabinovich subsequently generalised by the theorem of Hrushovski and Zilber classifying Zariski geometries. The preliminary version of the paper is on Boris Zilber's webpage.