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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.

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