On ramified torsion points on a curve with stable reduction over an absolutely unramified base

Abstract

Let p be an odd prime number, W an absolutely unramified p-adically complete discrete valuation ring with algebraically closed residue field, and X a curve of genus at least two over the field of fractions K of W. In the present paper, we study, under the assumption that X has stable reduction over W, torsion points on X, i.e., torsion points of the Jacobian variety J of X which lie on the image of the Albanese embedding X ↪→ J with respect to a K-rational point of X. A consequence of the main result of the present paper is that if, moreover, J has good reduction over W, then every torsion point on X is K-rational after multiplying p. This result is closely related to a conjecture of R. Coleman concerning the ramification of torsion points. For instance, this result leads us to a solution of the conjecture in the case where a given curve is hyperelliptic and of genus at least p.