The relative Bogomolov conjecture for fibered products of elliptic curves

Abstract

We deduce an analogue of the Bogomolov conjecture for non-degenerate subvarieties in fibered products of families of elliptic curves from the author’s recent theorem on equidistribution in families of abelian varieties. This generalizes results of DeMarco and Mavraki and improves certain results of Manin–Mumford type proven by Masser and Zannier to results of Bogomolov type, yielding the first results of this type for subvarieties of relative dimension > 1 {>1} in families of abelian varieties with trivial trace.

Dynamics on ℙ1: preperiodic points and pairwise stability

“…The full conjecture from [BFT18] is now a theorem: there exists a uniform bound so that every pair of (flexible) Lattès maps (in any choice of coordinates on ) will either share all of their preperiodic points or have at most in common. A proof was recently obtained by Poineau in [Poi22], and it can also be deduced from the ‘relative Bogomolov’ theorem proved by Kühne [Küh23] or the main theorem of [GGK21]. We present an alternate proof as a corollary to Theorem 1.3.…”

“…The latter case, where , will only hold if is a Lattès map arising as the quotient of a map on the same elliptic curve ; see § 2.3 for background on the preperiodic points of Lattès maps. The conjecture of Bogomolov, Fu, and Tschinkel is the case of Theorem 1.3 where is assumed to be a Lattès map, and it is stated here as Corollary 8.2; the conjecture has recently been established by Poineau [Poi22] and also follows by Kühne's work [Küh23] and a result of Gao, Ge, and Kühne [GGK21]. We also include a proof here to illustrate the differences in approach.…”