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The André-Oort conjecture for \(\mathcal A_g\). (English) Zbl 1415.11086

This paper gives an unconditional proof of the André-Oort conjecture for the moduli space \(\mathcal A_g\) of principally polarized abelian varieties, which asserts that an irreducible closed algebraic subvariety \(V\) of \(\mathcal A_g\) contains only finitely many maximal special subvarieties. (Here, a special subvariety is a subvariety which is the image of a connected Shimura variety under a morphism coming from a morphism of Shimura data.)
The main step towards proving this well-known conjecture carried out in the paper at hand is proving the following
Theorem (cf. Theorem 1.2). For given \(g\ge 1\) there exist constants \(\delta_g > 0\) and \(\gamma_g > 0\) depending only on \(g\) such that if \(E\) is a CM field of degree \(2g\), \(\Phi\) is a primitive CM type for \(E\), and \(A\) is an abelian variety of dimension \(g\) with endomorphism ring equal to the ring of integers of \(E\) and CM type \(\Phi\), then the field of moduli \(\mathbb Q(A)\) of \(A\) satisfies \[ \left| \mathbb Q(A) : \mathbb Q \right| \ge \gamma_g \left|\operatorname{Disc}(E)\right|^{\delta_g}. \]
The key ingredients in the proof are the “averaged Colmez conjecture” on the Faltings height of CM abelian varieties proved by Andreatta, Goren, Howard and Madapusi Pera and independently by Xuan and Zhang, and a theorem by Masser and Wüstholz bound the degrees of isogenies between such abelian varieties.
It was proved in [J. Pila and J. Tsimerman, Ann. Math. (2) 179, No. 2, 659–681 (2014; Zbl 1305.14020)] that this implies the André-Oort conjecture using, among other results, the theory of \(o\)-minimality and in particular a theorem by Pila and Wilkie on the growth of the number of points of bounded heights in definable sets. See Section 6 of the paper at hand for a sketch of the proof strategy.
Previously, the André-Oort conjecture had been proved under the assumption of the Generalized Riemann Hypothesis by Klingler, Ullmo and Yafaev ([B. Klingler and A. Yafaev, Ann. Math. (2) 180, No. 3, 867–925 (2014; Zbl 1377.11073); E. Ullmo and A. Yafaev, Ann. Math. (2) 180, No. 3, 823–865 (2014; Zbl 1328.11070)]) and also in the above-mentioned paper by Pila and Tsimerman.

MSC:

11G15 Complex multiplication and moduli of abelian varieties
11G18 Arithmetic aspects of modular and Shimura varieties
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References:

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