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A note on the ramification of torsion points lying on curves of genus at least two. (English) Zbl 1223.11075

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 \(p\) 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. The author shows that the coordinates of the torsion points of Jac(\(C\)) lying on \(C\) are contained in the unique tamely ramified quadratic extension of the field generated over \(K\) by the coordinates of the \(p\)-torsion points of Jac(\(C)\).

MSC:

11G20 Curves over finite and local fields
14H25 Arithmetic ground fields for curves
14H40 Jacobians, Prym varieties
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