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The rational points of a definable set. (English) Zbl 1217.11066
The authors obtain sharp upper bounds for the number of rational points of bounded height on certain analytic sets, by using real analytic geometry together with o-minimal structures from mathematical logic.
Let $$f:[0,1]\to {\mathbb R}$$ be a transcendental analytic function with graph $$X\subset [0,1]\times {\mathbb R}$$. Define the density function $$N(X,T)$$ of $$X$$ to be, for $$T>0$$, the number of $$P\in X({\mathbb Q})$$ with $$H(P)\leq T$$, where $$H(a/b)=\max\{|a|,b\}$$ for $$a/b\in{\mathbb Q}$$ with $$b>0$$ and $$\gcd(a,b)=1$$, and $$H(x_1,\dots,x_n)=\max_{1\leq i\leq n} H(x_i)$$ for $$(x_1,\dots,x_n)\in{\mathbb Q}^n$$.
Building upon a previous joint paper of the first author with E. Bombieri [Duke Math. J. 59, No. 2, 337–357 (1989; Zbl 0718.11048)] which answered a question by P. Sarnak on Betti numbers of abelian covers, the first author [Duke Math. J. 63, No. 2, 449–463 (1991; Zbl 0763.11025)] proved that for each $$\varepsilon>0$$, there is a constant $$c(X,\varepsilon)$$ such that $$N(X,T)\leq c(X,\varepsilon) T^\varepsilon$$. In a subsequent paper [Ann. Inst. Fourier 55, No. 5, 1501–1516 (2005; Zbl 1121.11032)], he extended this result to higher dimensional situation as follows. For $$X\subset {\mathbb R}^m$$, let $$X^{\mathrm{alg}}$$ denote the union of all connected semi-algebraic subsets of $$X$$ of positive dimension. Then, for any compact subanalytic set $$X\subset{\mathbb R}^n$$ of dimension $$2$$ and any $$\varepsilon>0$$, there is a constant $$c(X,\varepsilon)$$ such that $$N(X\setminus X^{\mathrm{alg}},T)\leq c(X,\varepsilon) T^\varepsilon$$.
In this paper, the authors prove more general statements by using the notion of o-minimal structure. They show that if $$X\subset {\mathbb R}^n$$ is definable in a given o-minimal structure over $${\mathbb R}$$, then for any $$\varepsilon>0$$ there is a constant $$c(X,\varepsilon)$$ such that $$N(X\setminus X^{\mathrm{alg}},T)\leq c(X,\varepsilon) T^\epsilon$$. A second version of the main result gives a similar upper bound for $$N(X\setminus X^{\mathrm{alg}},T)$$ when $$X$$ is a fibre of a definable family $$Z\subset {\mathbb R}^n\times{\mathbb R}^m$$, and the constant $$c$$ depends only on $$Z$$ and $$\varepsilon$$. The third and final version again deals with a definable family $$Z\subset {\mathbb R}^n\times{\mathbb R}^m$$. Given $$\varepsilon>0$$, there is a definable family $$W=W(Z,\varepsilon)\subset Z$$ and a constant $$c(Z,\varepsilon)$$ with the following property. Let $$y\in Y$$. Put $$X=X_{Z,y}$$ and $$X_\varepsilon=X_{W,y}$$. Then $$X_\varepsilon\subset X^{\mathrm{alg}}$$ and $$N(X\setminus X_\varepsilon,T)\leq c(Z,\varepsilon) T^\epsilon$$.
This paper has been the source of a number of important applications and developments. J. Pila and 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)] gave a new proof of the Manin–Mumford conjecture. D. Masser and U. Zannier [“Torsion anomalous points and families of elliptic curves”, Am. J. Math. 132, No. 6, 1677–1691 (2010; Zbl 1225.11078)] prove a new special case of the Zilber–Pink Conjecture. The first author [“Rational points of definable sets and results of André-Oort-Manin-Mumford type”, Int. Math. Res. Not. 2009, No. 13, 2476–2507 (2009; Zbl 1243.14021)] obtained new results related to the André–Oort–Manin–Mumford Conjecture.
The results of the paper under review were at the source of the resolution of the André-Oort Conjecture in the case of products of modular curves, giving the first unconditional proof of fundamental cases of these general conjectures beyond the original theorem of André concerning the product of two such curves: [J. Pila, “O-minimality and the André-Oort conjecture for $${\mathbb C}^n$$”, Ann. Math. (2) 173, 1779–1840 (2011)]. Further related papers are [J. Pila, “On the algebraic points of a definable set”, Sel. Math., New Ser. 15, No. 1, 151–170 (2009; Zbl 1218.11068)], [P. Habegger and J. Pila, “Some unlikely intersections beyond André-Oort”, Compositio Math., to appear] and [J. Pila and J. Tsimerman, “The André-Oort conjecture for the moduli space of Abelian surfaces”, arXiv:1106.4023v1].

##### MSC:
 11G99 Arithmetic algebraic geometry (Diophantine geometry) 11U09 Model theory (number-theoretic aspects) 03C64 Model theory of ordered structures; o-minimality
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