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

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

2009-02-24

arxiv.org/abs/0902.4161v1

Mathematics

Algebraic Geometry

Scientific paper

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 $\Jac(C)$.

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