Density of points and lower bound for the height (Q1434338)

Let \(K\supset k\) be two number fields, let \(A\) be an abelian variety defined over \(k\) and \(L\) a very ample line bundle over \(A\). Let \(V\) be an algebraic subvariety defined and irreducible over \(K\), such that there does not exist a finite union of translated of proper abelian subvarieties of \(A\) by torsion points containing \(V\). The author proves that for all \(\varepsilon>0\), the set of points \(x\in V(\overline{K})\) which are of infinite order in \(A\) modulo every proper abelian subvariety of \(A\), and of canonical height: \[ \widehat{h}_L(x)\leq \frac{\widehat{h}_L(V)}{\deg_L(V)}+\varepsilon \] is Zariski-dense in \(V\). This result can be used to prove two corollaries in the case \(A\) is of complex multiplication type; we suppose here that \(K=k\). The first corollary concerns the points of small canonical height lying over a variety \(V\) as above. Given such a \(V\), there exists a constant \(c>0\), depending on the couple \((A,L)\) only, such that the set of points \(x\in V(\overline{K})\) of canonical height \[ \widehat{h}_L(x)\leq\frac{c}{\delta} \left(\frac{ \log\log(3\delta)}{\log (3\delta)}\right)^{\kappa} \] is not Zariski-dense in \(V\). Here \(\delta\) is the obstruction index of \(V\) and \(\kappa>0\) is a constant, explicit and rather big, depending on \(\dim A\) only. The second corollary is a minoration of the essential minimum of a subvariety \(V\) of \(A\), defined over \(k\), which is not equal to a finite union of translated of proper abelian subvarieties of \(A\) by torsion points: \[ \mu^{\text{ess}}_L(V)\geq c\deg_L(V)^{\frac{1} {s-\dim(V)}}\cdot (\log(3 \deg_L(V)))^{-\kappa(s)}, \] where \(s\) is the dimension of the smallest algebraic subgroup of \(A\) containing \(V\) (the constant \(c\) is the same as above). The proof of these results heavily depends on techniques and results introduced and proved by \textit{F. Amoroso} and \textit{S. David} [Ramanujan J. 5, No.3, 237--246 (2001; Zbl 0996.11046)], \textit{S. David} and \textit{M. Hindry} [J. Reine Angew. Math. 529, 1--74 (2000; Zbl 0993.11034)] and \textit{S. Zhang} [J. Am. Math. Soc. 8, No. 1, 187--221 (1995; Zbl 0861.14018)]. More precisely, the proof of the main result of the revieved article follows the work of Amoroso-David, while the first corollary depends on the ``state of the art'' of the abelian Lehmer problem, which is very refined only if \(A\) is of C.M. type (work of David-Hindry), and the second corollary uses in addition a theorem of Zhang which says that: \[ \frac{\widehat{h}_L(V)}{\deg_L(V)}\geq \mu^{\text{ess}}_L(V)\geq \frac{\widehat{h}_L(V)}{\deg_L(V)(\dim V+1)}. \]