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Families of abelian varieties with many isogenous fibres. (English) Zbl 1349.14143
Let \(\mathcal A_g\) be the moduli space of principally polarized abelian varieties of dimension \(g\), and denote by \(\mathcal A_t\) the abelian variety corresponding to a point \(t\in\mathcal A_g(\mathbb C)\). For a fixed point \(s \in\mathcal A_g(\mathbb C)\), let \(\Lambda\) be the set of all \(t\in \mathcal A_g(\mathbb C)\) such that there exists an isogeny (not necessarily preserving the polarization) from \(\mathcal A_t\) to \(\mathcal A_s\). Let \(Z\) be an irreducible subvariety of \(\mathcal A_g\) such that \(Z \cap \Lambda\) is Zariski-dense in \(Z\). This paper is centered around the conjecture that under these hypotheses, \(Z\) is a weakly special subvariety of \(\mathcal A_g\). We recall that an irreducible subvariety of a Shimura variety is weakly special if and only if it is totally geodesic.
We now describe the main results. The conjecture stated above is shown to be a consequence of a conjecture of Zilber and Pink. The conjecture is shown to hold when \(Z\) is a curve. Without restriction on \(\dim Z\), the following is proved: there exists a special subvariety \(S \subset \mathcal A_g\) which is isomorphic to a product of Shimura varieties \(S_1 \times S_2\) with \(\dim S_1 > 0\), and such that \(Z = S_1 \times Z' \subset S\) for some closed subvariety \(Z' \subset S_2\).
The technical difficulties of this paper are due to the fact that the isogenies are not required to be compatible with the polarizations. In a recent preprint [M. Orr, “On compatibility between isogenies and polarizations of abelian varieties”, Preprint, arXiv:1506.04011], the author shows that any isogeny between abelian varieties induces a polarized isogeny between their fourth powers. This provides alternate (and perhaps more natural) proofs of the results of this paper.

MSC:
14K02 Isogeny
14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
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