Rational points on ramified covers of abelian varieties over torsion fields

It was recently shown by Corvaja, Demeio, Javanpeykar, Lombardo and Zannier that abelian varieties over number fields satisfy a certain Hilbert irreducibility type statement for rational points on ramified covers. I will motivate these notions and results and will report on subsequent joint work with Bary-Soroker and Petersen in which we extend these results to abelian varieties over certain infinite Galois extensions of number fields.