Last update: 12/08/02

Abstract

In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of Akio Tamagawa and the present authors for studying torsion points on curves with ordinary good or ordinary semistable reduction at a given prime. We also give new proofs of:

• The Manin-Mumford conjecture: There are only finitely many torsion points lying on a curve of genus at least 2 which is immersed in its Jacobian by an Albanese map.
• The Coleman-Kaskel-Ribet conjecture: If p≥23 is prime, then the only torsion points lying on the curve X₀(p), immersed in its Jacobian by a cuspidal embedding, are the cusps (together with the hyperelliptic branch points when X₀(p) is hyperelliptic and p ≠37).