Uniform bound on rational and algebraic points – UniBdPt

I propose to investigate the following long expected but widely open uniform bounds on rational and algebraic points.
(1) The number of rational points on a smooth projective curve of genus at least 2 defined over a number field of degree d is bounded above in terms of g, d and the Mordell-Weil rank.
(2) The number of algebraic torsion points on a smooth projective curve of genus g at least 2 defined over a number field of degree d is bounded above in terms of g and d.
(3) The number of rational torsion points on an abelian variety of dimension g defined over a number field of degree d is bounded above in terms of g and d.
Compared with existing results, the Faltings height is no longer involved in the bounds.
I expect to fully prove (1) and (2). For (3) it is high risk but I still expect to prove some new cases.
The proofs I propose are via Diophantine estimates. Functional transcendence and unlikely intersections on mixed Shimura varieties play important roles in the proofs. Hence as pre-requests and extensions of the three goals listed above, I will also continue investigating on transcendence theory and unlikely intersection theory as well as their potential other interesting applications in Diophantine geometry.