The rational points of a definable set

By Jonathan Pila, Alex Wilkie. 2006

246 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/86e1cb2fdec548d52c731336347b5443158852eb

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Definability of restricted theta functions and families of abelian varieties (2011, methodology): “[18] Jonathan Pila and Alex W. Wilkie, The rational points of a definable set, Duke Math.”
Some unlikely intersections beyond André–Oort (2011, methodology): “His approach relies on a counting result due to Wilkie and himself [38], and adopts a basic strategy proposed by Zannier in the context of the Manin-Mumford conjecture (see [39]); this strategy also plays a central role in the current article.”
O-minimality and the André-Oort conjecture for $\mathbb{C}^{n}$ (2011, methodology): “It was in fact this manuscript [81] that raised the questions about diophantine properties of smooth and analytic curves that led to the paper [16], whose ideas developed ultimately into [70].”
Ax-Lindemann for $\mathcal{A}_g$ (2014, methodology): “1 combines various arithmetic estimates with the Counting Theorem of Pila-Wilkie [29] and with the idea of Ullmo-Yafaev [40] to use hyperbolic volume at the boundary.”
Towards the Andr\'e-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points (2013, methodology): “Afterwards using tools of o-minimality and Pila-Wilkie’s counting theorem [51], Pila proved the André-Oort conjecture for A1 (product of modular curves) unconditionally [47].”
Counting Lattice Points and O-Minimal Structures (2012, methodology): “In Section 3 we collect some basic facts about o-minimal structures, as well as some deeper results like the cell-decomposition Theorem, the Reparametrization Lemma (originally due to Yomdin [36], [35], and Gromov [15, p.232], and refined by Pila and Wilkie [23]), and the existence of definable Skol”
Ax-Schanuel for Shimura varieties (2017, unknown): “It follows by the Pila-Wilkie theorem [35] that I contains irreducible real algebraic curves C containing arbitrarily many integer points, in particular, containing at least 2 integer points.”
Ax–Schanuel for the $j$-function (2014, methodology): “An application of the Counting Theorem of Pila-Wilkie [23] now shows that Z contains algebraic curves.”
O-minimality and certain atypical intersections (2014, methodology): “We will use the second-named author’s generalisation, stated below, of the Pila-Wilkie Theorem [34] to prove the following result for definable families.”
On the algebraic points of a definable set (2009, methodology): “With these denitions, the basic result of [ 16 ] is as follows.”