Colloquium: Tuesday, November 10, 2pm. Speaker: Assaf Hasson (Ben Gurion). Title: A curve and its Jacobian — on a theorem of Zilber and a theorem of Rabinovich.
Let C be a smooth projective curve (of genus g>1) over an algebraically closed field K. Let J(C) be the Jacobian of C. By Torelli’s theorem J(C), as a principally polarized Abelian variety determines C up to isomorphism. In 2010, Zilber proved that, in fact, much less information is needed in order to recover the isomorphism type of C. Indeed, fixing a point 0 in J(C) consider (J(C),0,+) as a pure Abelian group. Obviously, C is not encoded in this group. However, if we expand this group by the image of C under its canonical embedding in J(C) (i.e., we consider the structure (J(C), 0,+, C)) then any isomorphism between two such structures must arise from an isomorphism of the underlying algebraically closed fields, composed with a bijective isogeny of J(C). In particular, given a structure (J(C),0,+,C) as above, the ground field K can be reconstructed from the data.
The last statement is the key to Zilber’s proof, and it follows from a deep model theoretic result of E. Rabinovich. In the talk we will sketch Zilber’s proof, explain Rabinovich’s theorem, and how it is used by Zilber. If time allows I will give an outline of a new (and generalised) proof of Rabinovich’s theorem.
