Computing torsion points on curves

Abstract

Let X be a curve of genus g ≥ 2 over a field k of characteristic zero. Let X →A be an Albanese map associated to a point P0 on X. The Manin-Mumford conjecture, first proved by Raynaud, asserts that the set T of points in X(k) mapping to torsion points on A is finite. Using a p-adic approach, we develop an algorithm to compute T, and implement it in the case where k = Q, g = 2, and P0 is a Weierstrass point. Improved bounds on #T are also proved: for instance, in the context of the previous sentence, if in addition X has good reduction at a prime p > 5, then #T < 2p3 + 2p2 + 2p + 8. © A K Peters, Ltd.