  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
A. 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: (1) The Manin-Mumford conjecture: There are only finitely
many torsion points lying on a curve of genus at least 2 embedded in its
Jacobian by an Albanese map; and (2) The Coleman-Kaskel-Ribet conjecture: If p
is a prime number which is at least 23, then the only torsion points lying on
the curve X_0(p), embedded in its Jacobian by a cuspidal embedding, are the
cusps (together with the hyperelliptic branch points when X_0(p) is
hyperelliptic and p is not 37). In an effort to make the exposition as useful
as possible, we provide references for all of the facts about modular curves
which are needed for our discussion.
