Special points on fibered powers of elliptic surfaces (Q2872166)

The present paper concern special points on a certain class of abelian schemes over \(\bar{\mathbb Q}\). Let \(S\) be a non-singular quasi-projective curve defined over \(\bar{\mathbb Q}\), and \(\mathcal E \to S\) an elliptic surface, i.e., an abelian scheme over \(S\) whose (smooth) fibers are elliptic curves, again defined over \(\bar{\mathbb Q}\). Consider for an integer \(g\geq 1\) the \(g\)-fold self fiber product NEWLINE\[NEWLINE \mathcal A = \mathcal E \times_S\cdots\times_S \mathcal E. NEWLINE\]NEWLINE This is again an abelian scheme over \(S\) defined over \(\bar{\mathbb Q}\).NEWLINENEWLINEThe paper's main result blends the Manin-Mumford conjecture and the André-Oort conjecture by characterising the subvarieties of \(\mathcal A\) (over the base change to \(\mathbb C\)!) which contain a Zariski dense set of points appearing as torsion points on CM fibers. More precisely, assuming that \(\mathcal A\) is not isotrivial, the author proves that an irreducible closed subvariety of \(\mathcal A\) over \(\mathbb C\) contains a Zariski dense set of special points if and only if it is special itself.NEWLINENEWLINEAs a key ingredient for the proof, the author develops a new height inequality on \(\mathcal A\) which generalizes work of Silverman for elliptic curves. The height inequality is also used to give an alternative proof of the Bogomolov Conjecture for products of elliptic curves over the function field of a curve (as proved by Gubler).