Computing Torsion Points on Curves

Bjorn Poonen
2001 Experimental Mathematics  
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 P 0 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 P 0 is a Weierstrass point. Improved bounds on #T are also proved: for instance, in the context of the previous
more » ... xt of the previous sentence, if moreover X has good reduction at a prime p ≥ 5, then #T ≤ 2p 3 + 2p 2 + 2p + 8.
doi:10.1080/10586458.2001.10504462 fatcat:2tdcqojqajeyzoer6juom5vpte