Diophantine Geometry: towards the ultimate Bogomolov conjecture

This project proposes research with a view towards extensions of the Bogomolov conjecture beyond the original setting of abelian varieties.

In the past decade, there have been some indications that this may be possible: (a) Masser and Zannier have proven ""relative'' analogues of the Manin-Mumford conjecture in various families of abelian varieties, (b) DeMarco and Mavraki have shown a ""relative'' analogue of the Bogomolov conjecture for sections in a fibered product of elliptic families, and (c) Ghioca, Nguyen, and Ye have proven a ""dynamical'' Bogomolov conjecture for split rational maps.

A prominent tool in almost all proofs of the Bogomolov conjecture are equidistribution techniques (i.e., Yuan's equidistribution theorem). However, there are two problems with this approach when it comes to ""relative'' generalizations.

First, the Néron-Tate local height in families of abelian varieties exhibits b-singularities nearby degenerate fibers, preventing a direct use of Yuan's theorem if the family has degenerate fibers.

Recently, I have overcome these problems and proven a satisfactory analogue of the equidistribution conjecture in families of abelian varieties over a base curve.

Part of the research proposed here is to generalize and exploit this result further.Second, equidistribution techniques usually fall short of ""relative'' Bogomolov-type results -- in stark contrast to the case of abelian varieties. Similar problems arise in the ""dynamical"" setting, indicating a profound conceptional obstacle.

For this reason, it is proposed here to adapt a method of David and Philippon, who gave an equidistribution-free direct proof of the Bogomolov conjecture for abelian varieties, to the relative setting.

Such a method, if successful, should shed some light on an ""ultimate'' Bogomolov conjecture encompassing virtually all the Bogomolov-type results known up to the present day.