A note on the ramification of torsion points lying on curves of genus at least two

TL;DR

This paper investigates the ramification properties of torsion points on algebraic curves of genus at least two, showing they lie in a specific quadratic extension related to the Jacobian's torsion points.

Contribution

It establishes that torsion points on such curves over certain fields are contained in a uniquely defined moderately ramified quadratic extension.

Findings

Torsion points on genus ≥ 2 curves lie in a specific quadratic extension.

Coordinates of torsion points are contained in a moderately ramified quadratic extension.

The result applies under semi-stable reduction and certain characteristic assumptions.

Abstract

Let $C$ be a curve of genus $g\geqslant 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $\char(K)=0$ and that the characteristic of the residue field is not 2. Suppose that the Jacobian $\Jac(C)$ has semi-stable reduction over $R$. Embed $C$ in $\Jac(C)$ using a $K$-rational point. We show that the coordinates of the torsion points lying on $C$ lie in the unique moderately ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion points on $\Jac(C)$.