Duke Mathematical Journal

Ramification of torsion points on curves with ordinary semistable Jacobian varieties

Akio Tamagawa

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Let X be a proper, smooth, geometrically connected curve of genus g≥2 over a p-adically complete discrete valuation field K. By the Albanese morphism withrespect to a given K-rational point, the curve X can be embedded into its Jacobian variety J. Then, assuming that J has ordinary semistable reduction, we prove that the inertia group of K acts trivially on the set of torsion points of J which lie on X, under certain mild conditions. As an application, we prove that the modular curve X0(N) (N: a prime number greater than or equal to 23), embedded into its Jacobian variety by using a cusp, contains no torsion points other than the cusps (resp., the cusps and the Weierstrass points), if N∉{23,29,31,41,47,59,71} (resp., N∈{23,29,31,41,47,59,71}). This affirmatively answers a question posed by R. Coleman, B. Kaskel, and K. Ribet [CKR].

Article information

Duke Math. J., Volume 106, Number 2 (2001), 281-319.

First available in Project Euclid: 13 August 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22]


Tamagawa, Akio. Ramification of torsion points on curves with ordinary semistable Jacobian varieties. Duke Math. J. 106 (2001), no. 2, 281--319. doi:10.1215/S0012-7094-01-10623-6. https://projecteuclid.org/euclid.dmj/1092403917

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