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integral lattice points, and that the exponent and constant are best possible. However, Swinnerton–Dyer [10] showed that the preceding result can be substantially improved if we start with a fixed,… (More)

LetX ⊂ R be a set that is definable in an o-minimal structure overR. This paper shows that, in a suitable sense, there are very few rational points ofX that do not lie on some connected semialgebraic… (More)

- Jonathan Pila
- 2011

We give an unconditional proof of the André-Oort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the Manin-Mumford conjecture for arbitrary… (More)

We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points… (More)

- Jonathan Pila
- 1995

- Jonathan Pila
- 2004

Let Ω ⊂ R be a compact subanalytic set of dimension 2 and t ≥ 1. This paper gives an upper bound as t → ∞ for the number of integer points on the homothetic dilation tΩ of Ω that do not reside on any… (More)

- Jonathan Pila
- 2009

- Jonathan Pila
- 1990

We give a generalization to Abelian varieties over finite fields of the algorithm of Schoof for elliptic curves. Schoof showed that for an elliptic curve E over F , given by a Weierstrass equation,… (More)