Jonathan Pila

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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 as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety(More)
This paper presents an algorithm that, given an integer n > 1, finds the largest integer k such that n is a kth power. A previous algorithm by the first author took time b1+o(1) where b = lg n; more precisely, time b exp(O( √ lg b lg lg b)); conjecturally, time b(lg b)O(1). The new algorithm takes time b(lg b)O(1). It relies on relatively complicated(More)
In a recent paper I established an analogue of the Lindemann-Weierstrass part of Ax-Schanuel for the elliptic modular function. Here I extend this to include its first and second derivatives. A generalisation is given that includes exponential and Weierstrass elliptic functions as well. 2000 Mathematics Subject Classification: 11G18, 03C64
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