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Ramification of torsion points on curves with ordinary semistable Jacobian varieties. (English) Zbl 1010.14007

Let \(K\) be a complete discrete valuation field of characteristic zero and of residue characteristic \(p>0\). We denote by \(I_K\) the absolute inertia group of \(K\). We consider a proper smooth, geometrically connected curve \(X\) of genus \(g\geq 2\) and we denote by \(J\) its Jacobian variety. By the Albanese morphism with respect to a given \(K\)-rational point \(P\), the curve \(X\) can be embedded into \(J\). We put \(J_P=X(\overline K)\cap J_{\text{tors}} (\overline K)\). In the paper under review, the authors, assuming that \(J\) has ordinary semistable reduction, prove that \(I_K\) acts trivially on \(J_P\), under certain mild conditions.
An interesting application of this result is given in case where \(X\) is the modular curve \(X_0(N)\), where \(N\) is a prime number \(\geq 23\) and \(P\) is one of the two cusps of \(X_0(N)\). More precisely, the authors prove, under the above conditions, that \(J_P=\){the two cusps}, if \(N \notin \{23,29,31, 41,47,59,71\}\) and \(J_P=\){the two cusps}\(\cup W\), otherwise, where \(W\) denotes the set of Weierstrass points of \(X_0(N)\).

MSC:

14H40 Jacobians, Prym varieties
14H55 Riemann surfaces; Weierstrass points; gap sequences
11G18 Arithmetic aspects of modular and Shimura varieties
14H25 Arithmetic ground fields for curves
14G35 Modular and Shimura varieties
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