Ramification of torsion points on a curve with superspecial reduction over an absolutely unramified base

Abstract

Let $p$ be a prime number, $W$ an absolutely unramified $p$-adically complete discrete valuation ring with perfect residue field, and $X$ a curve over the field of fractions of $W$ of genus greater than one. In the present paper, we study the ramification of torsion points on the curve $X$. A consequence of the main result of the present paper is nonexistence of ramified torsion point on $X$ in the case where $p$ is greater than three, the Jacobian variety $J$ of $X$ has good reduction over $W$, and the special fiber of the good model of $J$ is superspecial. This consequence generalizes a theorem proved by Coleman.