The Relative Bogomolov Conjecture for Fibered Products of Elliptic Curves

TL;DR

This paper extends the Bogomolov conjecture to non-degenerate subvarieties in fibered products of elliptic curves, using equidistribution theorems, and generalizes previous results to higher-dimensional subvarieties in abelian families.

Contribution

It introduces a new Bogomolov-type result for higher-dimensional subvarieties in fibered elliptic curves, building on recent equidistribution theorems.

Findings

Establishes Bogomolov-type bounds for subvarieties of fibered elliptic curves.

Generalizes previous Manin-Mumford results to higher dimensions.

First such results for subvarieties of relative dimension >1 in trivial trace abelian families.

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 $>1$ in families of abelian varieties with trivial trace.