Rational points on analytic varieties (Q2352770)

This very readable survey gives an introduction to the work of Pila (and others), on counting rational points on analytic varieties, and describes the basic applications. Let \(X\subset\mathbb{R}^m\) be a compact real-analytic transcendental manifold, and let \(X(N)\) be the set of rational points \((a_1/q_1,\ldots,a_m/q_m)\in X\) (with \(a_i\in\mathbb{Z}\), \(q_i\in\mathbb{N}\)) such that \(\max(|a_i|,q_i)\leq N\) for each \(i\). The basic result in the area says that if one removes points that lie on algebraic subvarieties of \(X\) there are \(O_{\varepsilon,X}(N^{\varepsilon})\) points left in \(X(N)\), for any fixed \(\varepsilon>0\). The paper discusses the background to this result, and describes the proof first in the case \(m=2\) and then for \(m=3\), building on the seminal papers by \textit{E. Bombieri} and \textit{J. Pila} [Duke Math. J. 59, No. 2, 337--357 (1989; Zbl 0718.11048)] and by \textit{J. Pila} [Ann. Inst. Fourier 55, No. 5, 1501--1516 (2005; Zbl 1121.11032)]. The extension to sets definable in an o-minimal structure is then described. The survey concludes by showing how these results may be applied to the Manin-Mumford and André-Oort conjectures.