Katz and Oort raised the following question: Given an algebraically closed field k, and a positive integer g>3, does there exist an abelian variety over k not isogenous to a Jacobian over k? There has been much progress on this question, with several proofs now existing over ℚ. We discuss recent work with Ananth Shankar, answering this question in the affirmative over 𝔽q(T). Our method introduces new types of local obstructions, and can be used to give another proof over ℚ.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
