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 {\it 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 {\it stable reduction} over $W$, {\it torsion points} on $X$, i.e., torsion points of the Jacobian variety $J$ of $X$ which lie on the image of the Albanese embedding $X\hookrightarrow J$ 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 {\it $K$-rational} {\it after multiplying $p$}. This result is closely related to a conjecture of {\it 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 $p$.