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

Damian Rössler[1]

  • [1] Département de Mathématiques Bâtiment 425 Faculté des Sciences d’Orsay Université Paris-Sud 91405 Orsay Cedex, FRANCE

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 2, page 475-481
  • ISSN: 1246-7405

Abstract

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Let C be a curve of genus g ≥ 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. We show that the coordinates of the torsion points lying on C lie in the unique tamely ramified quadratic extension of the field generated over K by the coordinates of the p -torsion points on Jac ( C ) .

How to cite

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Rössler, Damian. "A note on the ramification of torsion points lying on curves of genus at least two." Journal de Théorie des Nombres de Bordeaux 22.2 (2010): 475-481. <http://eudml.org/doc/116415>.

@article{Rössler2010,
abstract = {Let $C$ be a curve of genus $g\ge 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $\operatorname\{char\}(K)=0$ and that the characteristic $p$ of the residue field is not $2$. Suppose that the Jacobian $\operatorname\{Jac\}(C)$ has semi-stable reduction over $R$. Embed $C$ in $\operatorname\{Jac\}(C)$ using a $K$-rational point. We show that the coordinates of the torsion points lying on $C$ lie in the unique tamely ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion points on $\operatorname\{Jac\}(C)$.},
affiliation = {Département de Mathématiques Bâtiment 425 Faculté des Sciences d’Orsay Université Paris-Sud 91405 Orsay Cedex, FRANCE},
author = {Rössler, Damian},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {curves of genus at least two; jacobian; torsion points},
language = {eng},
number = {2},
pages = {475-481},
publisher = {Université Bordeaux 1},
title = {A note on the ramification of torsion points lying on curves of genus at least two},
url = {http://eudml.org/doc/116415},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Rössler, Damian
TI - A note on the ramification of torsion points lying on curves of genus at least two
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 2
SP - 475
EP - 481
AB - Let $C$ be a curve of genus $g\ge 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $\operatorname{char}(K)=0$ and that the characteristic $p$ of the residue field is not $2$. Suppose that the Jacobian $\operatorname{Jac}(C)$ has semi-stable reduction over $R$. Embed $C$ in $\operatorname{Jac}(C)$ using a $K$-rational point. We show that the coordinates of the torsion points lying on $C$ lie in the unique tamely ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion points on $\operatorname{Jac}(C)$.
LA - eng
KW - curves of genus at least two; jacobian; torsion points
UR - http://eudml.org/doc/116415
ER -

References

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  1. Baker, M; Poonen, B., Torsion packets on curves. Compositio Math. 127 (2001), 109–116. Zbl0987.14020MR1832989
  2. Baker, M.; Ribet, K., Galois theory and torsion points on curves. J. Théor. Nombres Bordeaux 15 (2003), 11–32. Zbl1065.11045MR2018998
  3. Boxall, J., Autour d’un problème de Coleman. C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 1063–1066. Zbl0803.14024MR1191490
  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. Zbl0840.14029MR1345173
  5. Coleman, R., Ramified torsion points on curves. Duke Math. J. 54 (1987), 615–640. Zbl0626.14022MR899407
  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. Zbl0237.00013MR354656
  7. Hartshorne, R., Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, 1977. Zbl0367.14001MR463157
  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. Zbl1098.14030MR2176757
  9. Serre, J.-P., Local fields. Graduate Texts in Mathematics 67. Springer-Verlag, 1979. Duke Math. J. 106 (2001), 281–319. Zbl0423.12016MR554237
  10. Tamagawa, A., Ramification of torsion points on curves with ordinary semistable Jacobian varieties. Duke Math. J. 106 (2001), 281–319. Zbl1010.14007MR1813433

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