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 Po on X. The Manin–Mumford conjecture, first proved by Raynaud, asserts that the set T of points in X() 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. lmproved 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.