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 products of el-liptic curves defined over Q as well as Lang's conjecture for torsion points in powers of the multiplicative group. The second includes the… (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 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 b 1+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.
Let X ⊂ R 2 be the graph of a pfaffian function f in the sense of Khovanskii. Suppose that X is nonalgebraic. This note gives an estimate for the number of rational points on X of height ≤ H; the estimate is uniform in the order and degree of f .