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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