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(\(C)\).


11G20 Curves over finite and local fields
14H25 Arithmetic ground fields for curves
14H40 Jacobians, Prym varieties
Full Text: DOI arXiv EuDML


[1] Baker, M; Poonen, B., Torsion packets on curves. Compositio Math. 127 (2001), 109-116. · Zbl 0987.14020
[2] Baker, M.; Ribet, K., Galois theory and torsion points on curves. J. Théor. Nombres Bordeaux 15 (2003), 11-32. · Zbl 1065.11045
[3] Boxall, J., Autour d’un problème de Coleman. C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 1063-1066. · Zbl 0803.14024
[4] Boxall, J., Sous-variétés algébriques de variétés semi-abéliennes sur un corps fini. Number theory 1992-1993. London Math. Soc. Lecture Note Ser. 215, 69-80. Cambridge Univ. Press, 1995. · Zbl 0840.14029
[5] Coleman, R., Ramified torsion points on curves. Duke Math. J. 54 (1987), 615-640. · Zbl 0626.14022
[6] Groupes de monodromie en géométrie algébrique. I. Lecture Notes in Mathematics, Vol. 288. Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7 I); Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. Springer-Verlag, Berlin, 1972. · Zbl 0237.00013
[7] Hartshorne, R., Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, 1977. · Zbl 0367.14001
[8] Rössler, D., A note on the Manin-Mumford conjecture. Number fields and function fields—two parallel worlds. Progr. Math. 239, 311-318. Birkhäuser, 2005. · Zbl 1098.14030
[9] Serre, J.-P., Local fields. Graduate Texts in Mathematics 67. Springer-Verlag, 1979. Duke Math. J. 106 (2001), 281-319. · Zbl 0423.12016
[10] Tamagawa, A., Ramification of torsion points on curves with ordinary semistable Jacobian varieties. Duke Math. J. 106 (2001), 281-319. · Zbl 1010.14007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.