Towards an unconditional proof of the André-Oort Conjecture and surrounding problems

Abstract

We take steps towards carrying out Pila's strategy for proving the André-Oort conjecture unconditionally. Specifically, we prove lower bounds for galois orbits of special points in the [special characters omitted] for g ≤ 6. We then (joint with J.Pila) carry out the rest of Pila's program for [special characters omitted] and obtain an unconditional proof of André-Oort in this case. We also settle a conjecture of Nicholas Katz and Frans Oort by proving that for every g ≥ 4, there exists an abelian variety of dimension g over [special characters omitted] that's not isogenous to any Jacobian.