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Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension. (English) Zbl 1176.14008
Consider first an irreducible closed subvariety $$X$$ of a semiabelian variety $$S$$ over $${\mathbb C}$$, and remove from it all translated subgroups of $$S$$ that have large intersection with $$X$$. The assertion in this paper is that if $$S=A$$ is an abelian variety, and if $$A$$ and $$X$$ are both defined over $$\overline{\mathbb Q}$$, then the Néron-Tate height is bounded on $$\overline{\mathbb Q}$$-points that survive this process and are in a suitable sense close to a subgroup of small dimension.
Here is a more precise statement. Let $$n=\dim S$$, and for $$s\in{\mathbb N}$$, define $$X^{{\text{oa,}[s]}}$$ to be the complement of every subvariety $$Y\subset X$$ of positive dimension such that $$Y$$ is after translation also a subvariety of codimension $$<n-s$$ in an algebraic subgroup of $$S$$. On an abelian variety $$A$$, with $$A$$ and $$X$$ both defined over $${\overline{\mathbb Q}}$$, let $$\hat h$$ denote the Néron-Tate height (with respect to some symmetric ample line bundle) and for $$\Sigma\subset A({\mathbb C})$$ and $$\epsilon>0$$, put ${\mathcal C}(\sigma,\epsilon)=\{a+b\mid a,\;b\in A({\overline{\mathbb Q}}), \;a\in\Sigma,\;\hat h(b)\leq\epsilon(1+\hat h(a))\}.$ For $$s\in{\mathbb N}$$, denote by $$A^{[s]}$$ the union of all algebraic subgroups of $$A$$ of codimension at least  $$s$$. Then the result is that for some $$\epsilon>0$$ the height is bounded above on $$X^{{\text{oa,}}[s]}\cap {\mathcal C}(A^{[s]},\epsilon)$$.
This result is the analogue for abelian varieties over $$\overline{\mathbb Q}$$ of the bounded height conjecture of Bombieri, Masser and Zannier for $${\mathbb G}_m^n$$. If $$A$$ has complex multiplication there are some illuminating corollaries. Firstly, $$X^{{\text{oa,}}[s]}({\overline{\mathbb Q}})\cap A^{[1+s]}$$ is finite. If, furthermore, any surjective homomorphism $$\phi: A\to B$$ to an abelian variety has $$\phi(X)$$ as big as possible, i.e. $$\dim \phi(X)=\min(\dim X, \dim B)$$, then $$X^{{\text{oa,}}[\dim X]}\neq \emptyset$$ and $$X({\mathbb Q})\cap A^{[1+\dim X]}$$ is not Zariski dense in $$X$$.
This last statement is a part of the following conjecture: if $$X$$ is an irreducible subvariety of a semiabelian variety $$S$$ (both defined over $${\mathbb C}$$), not contained in a proper subgroup of $$S$$, then $$X({\mathbb C})\cap S^{[1+\dim X]}$$ is not Zariski dense in $$X$$. That would follow from a conjecture of Pink and would imply the Mordell-Lang conjecture: it would also follow from another conjecture of Zilber.
The proof is complicated. It depends on bounding the height of $$\psi(p)$$ for $$p\in X^{{\text{oa,}}[s]}$$ and suitable $$\psi: A\to B$$, and doing so in a rather precise way. Other essential ingredients are a result of Ax bounding the dimensions of the subgroups of $$A$$ generated by certain subsets, and generalities about abelian varieties such as the theorem of the cube. Because the latter uses completeness in an essential way, a proof for semiabelian varieties would have to be significantly different.

##### MSC:
 14K15 Arithmetic ground fields for abelian varieties 11G50 Heights 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
##### Keywords:
abelian variety; height; Diophantine approximation
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##### References:
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