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$$o$$-minimality and the André-Oort conjecture for $$\mathbb{C}^{n}$$. (English) Zbl 1243.14022
This paper provides new unconditional results related to conjectures by André, Oort, Manin, Mumford, Mordell, Lang and Bogomolov by using the theory of $$o$$-minimal structures from Model Theory. Let $$n$$, $$m$$, $$\ell$$ be nonnegative integers, $$\Gamma_1,\dots,\Gamma_n$$ be congruence subgroups of $$\text{SL}_2(\mathbb Z)$$, $$Y_i=\Gamma_i\backslash \mathbb H$$ ($$i=1,\dots,n$$) the corresponding modular curves, $$E_1,\dots,E_m$$ elliptic curves defined over the field $${\overline{\mathbb Q}}$$ of algebraic numbers, $$\text{G}$$ the multiplicative group $$\text{G}_m(\mathbb C)$$, $$X$$ the product $$Y_1 \times \cdots \times Y_n\times E_1 \times\cdots \times E_m \times \text{G}^\ell$$ and $$V$$ a subvariety of $$X$$.
One of the main results of this paper is that $$V$$ contains only a finite number of maximal special subvarieties. An equivalent statement is the following: if $$\Sigma$$ is a set of special points of $$X$$ with Zariski closure $$V_\Sigma$$, then the irreducible components of $$V_\Sigma$$ are special subvarieties. For the special case $$n=\ell=0$$ with $$X=Y_1 \times \cdots \times Y_n$$, this solves the André–Oort conjecture for the product of modular curves (the general case of this conjecture deals with Shimura varieties). For $$n=0$$ and $$X=E_1 \times\cdots \times E_m\times \text{G}^\ell$$, the result is a special case of a result of M. Hindry which includes the case $$n=\ell=0$$, $$X=E_1 \times\cdots \times E_m$$ (result of M. Raynaud on the Manin–Mumford conjecture) and $$m=n=0$$, $$X=\text{G}^\ell$$ (result of M. Laurent on a conjecture of Lang on the intersection of a subvariety of a semi-algebraic variety with the set of division points of a finitely generated group of rational points).
The strategy of the present paper was proposed by U. Zannier; it has its source in an unpublished manuscript by P. Sarnak [“Torsion points on varieties and homology of abelian covers”, unpublished manuscript (1998)]. See also P. Sarnak and S. Adams [Isr. J. Math. 88, No. 1–3, 31–72 (1994; Zbl 0843.11027)]. Sarnak’s work led to the paper by E. Bombieri and J. Pila [Duke Math. J. 59, 337–357 (1989; Zbl 0718.11048)], which was then developed by the author and A. J. Wilkie [Duke Math. J. 133, 591–616 (2006; Zbl 1217.11066)]. Three recent papers follow this strategy and involve proofs along similar lines. In a joint paper with U. Zannier [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 19, No. 2, 149–162 (2008; Zbl 1164.11029)], the author gave a new proof of a Manin–Mumford Conjecture. D. Masser and U. Zannier [Am. J. Math. 132, No. 6, 1677–1691 (2010; Zbl 1225.11078)] gave a proof of a special but new case of Pink’s relative Manin–Mumford Conjecture. And the author [Int. Math. Res. Not. 2009, No. 13, 2476–2507 (2009; Zbl 1243.14021)] gave new proofs of certain known results of André–Oort–Manin–Mumford type, including the cases ($$n=2$$, $$m=\ell=0$$) and ($$n=m=1$$, $$\ell=0$$).

##### MSC:
 14G05 Rational points 03C64 Model theory of ordered structures; o-minimality 11D45 Counting solutions of Diophantine equations 11G18 Arithmetic aspects of modular and Shimura varieties 11J95 Results involving abelian varieties 11U09 Model theory (number-theoretic aspects) 14K15 Arithmetic ground fields for abelian varieties
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