The André-Oort conjecture for \(\mathcal A_g\) (Q1698037): Difference between revisions

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.) \par The main step towards proving this well-known conjecture carried out in the paper at hand is proving the following \par 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 \[ \left| \mathbb Q(A) : \mathbb Q \right| \ge \gamma_g \left|\operatorname{Disc}(E)\right|^{\delta_g}. \] \par 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. \par It was proved in [\textit{J. Pila} and \textit{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. \par Previously, the André-Oort conjecture had been proved under the assumption of the Generalized Riemann Hypothesis by Klingler, Ullmo and Yafaev ([\textit{B. Klingler} and \textit{A. Yafaev}, Ann. Math. (2) 180, No. 3, 867--925 (2014; Zbl 1377.11073); \textit{E. Ullmo} and \textit{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.