Computing Torsion Points on Curves

Abstract

Let $X$ be a curve of genus $g \ge 2$ over a field $k$ of characteristic zero. Let $X\box0 A$ be an Albanese map associated to a point $P_0$ on $X$. The Manin--Mumford conjecture, first proved by Raynaud, asserts that the set $T$ of points in $X(\kbar)$ 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 $k=\funnyQ$, $g=2$, and $P_0$ 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 $p \ge 5$, then $\#T \le\ 2 p^3 + 2 p^2 + 2p + 8$.