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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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