In the early 1970s, Borel and Tits proved that if two simple algebraic groups overs algebraically closed fields can be shown to be abstractly isomorphic as groups, then the isomorphism yields an isomorphism of their underlying algebraically closed fields which in turn yields an isogeny, or 'near isomorphism', between the algebraic groups. What this means is that the algebra determines the geometry for simple algebraic groups: from the group structure, one can recover most, if not all, of the geometric structure. This same ideology has been prevalent in various parts of algebraic geometry, and I will present one such instance of its appearance within anabelian geometry.

In 2008, Bogomolov, Tschinkel, and Korotiaev prove that if C, C' are two smooth projective bar{F_p}-curves of genus two or greater, the existence of an abstract group isomorphism between their associated Jacobian abelian varieties yields an isogeny between J and J'. They further conjectured that under the aforementioned assumptions, C and C' are isomorphic as varieties modulo a Frobenius twisting. I will show you a proof of their conjecture given by Boris Zilber via E.D. Rabinovich's theorem: assume that M is a strongly minimal structure definable in an algebraically closed field K, and assume that the universe of M is some rational K-curve. Then if M is not locally modular there exists a field F interpretable within M that is isomorphic to K.