A Curve and its Abstract Jacobian
Boris Zilber · International Mathematics Research Notices · 2012
Let C(K) be the K-points of a smooth projective curve C of genus g>1 and J(K) its Jacobian. Fixing a point on the curve, one has a canonical embedding of C(K) into J(K) with the point identified as 0 of the group. Consider 〈J(K);+,C(K)〉 as an abstract structure of a group with a distinguished subset. Using model theory, we prove that, for K algebraically closed, one can recover from these data the field K and the curve C, up to isomorphisms of fields. If C is defined over a finite field, then C is determined as an algebraic curve uniquely up to Frobenius morphisms. This, in particular, proves a conjecture posed by F. Bogomolov, M. Korotaev, and Yu. Tschinkel.