Abelian varieties not isogenous to Jacobians

Joint IAS/Princeton University Number Theory Seminar Topic: Abelian varieties not isogenous to Jacobians Speaker: Jacob Tsimerman Affiliation: University of Toronto Date: December 01, 2021 Katz and Oort raised the following question: Given an algebraically closed field k, and a positive integer g greater than 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⎯⎯⎯⎯⎯.