Osaka Journal of Mathematics

On ramified torsion points on a curve with stable reduction over an absolutely unramified base

Yuichiro Hoshi

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Let $p$ be an odd prime number, $W$ an {\it absolutely unramified} $p$-adically complete discrete valuation ring with algebraically closed residue field, and $X$ a curve of genus at least two over the field of fractions $K$ of $W$. In the present paper, we study, under the assumption that $X$ has {\it stable reduction} over $W$, {\it torsion points} on $X$, i.e., torsion points of the Jacobian variety $J$ of $X$ which lie on the image of the Albanese embedding $X\hookrightarrow J$ with respect to a $K$-rational point of $X$. A consequence of the main result of the present paper is that if, moreover, $J$ has good reduction over $W$, then every torsion point on $X$ is {\it $K$-rational} {\it after multiplying $p$}. This result is closely related to a conjecture of {\it R. Coleman} concerning the ramification of torsion points. For instance, this result leads us to a solution of the conjecture in the case where a given curve is hyperelliptic and of genus at least $p$.

Article information

Osaka J. Math., Volume 54, Number 4 (2017), 767-787.

First available in Project Euclid: 20 October 2017

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Zentralblatt MATH identifier

Primary: 14H25: Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx]
Secondary: 11G20: Curves over finite and local fields [See also 14H25] 14H40: Jacobians, Prym varieties [See also 32G20] 14H55: Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx] 14L15: Group schemes 11S15: Ramification and extension theory


Hoshi, Yuichiro. On ramified torsion points on a curve with stable reduction over an absolutely unramified base. Osaka J. Math. 54 (2017), no. 4, 767--787.

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  • E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris: Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften Springer-Verlag, New York, 1985.
  • M. Baker: Torsion points on modular curves, Ph.D. thesis, University of California, Berkeley, 1999.
  • S. Bosch, W. Lütkebohmert and M. Raynaud: Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1990.
  • R.F. Coleman: Ramified torsion points on curves, Duke Math. J. 54 (1987), 615–640.
  • R.F. Coleman, B. Kaskel and K.A. Ribet: Torsion points on $X_0(N)$, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), 27–49, Proc. Sympos. Pure Math., 66, Part 1, Amer. Math. Soc., Providence, RI, 1999.
  • G. Faltings and C.-L. Chai: Degeneration of abelian varieties. With an appendix by David Mumford, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1990.
  • J.-M. Fontaine: Formes différentielles et modules de Tate des variétés abéliennes sur les corps locaux, Invent. Math. 65 (1981/82), 379–409.
  • J.-M. Fontaine: Il n'y a pas de variété abélienne sur $\mathbb Z$, Invent. Math. 81 (1985), 515–538.
  • Y. Hoshi: Tame-blind extension of morphisms of truncated Barsotti-Tate group schemes, J. Math. Sci. Univ. Tokyo 16 (2009), 23–54.
  • J.S. Milne: Arithmetic duality theorems, Second edition. BookSurge, LLC, Charleston, SC, 2006.
  • M. Raynaud: Schémas en groupes de type $(p , \dots , p)$, Bull. Soc. Math. France 102 (1974), 241–280.
  • D. Rössler: A note on the ramification of torsion points lying on curves of genus at least two, J. Théor. Nombres Bordeaux 22 (2010), 475–481.
  • A. Tamagawa: Ramification of torsion points on curves with ordinary semistable Jacobian varieties, Duke Math. J. 106 (2001), 281–319.