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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The strategy of the present paper was proposed by U. Zannier; it has its source in an unpublished manuscript by P. Sarnak [“Torsion points on varieties and homology of abelian covers”, unpublished manuscript (1998)].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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