Ramification of torsion points on curves with ordinary semistable Jacobian varieties
type: Item
instance of: scholarly article
title: Ramification of torsion points on curves with ordinary semistable Jacobian varieties
zbMATH Open document ID: 1010.14007
DOI: 10.1215/S0012-7094-01-10623-6
author: Akio Tamagawa
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: Let \(K\) be a complete discrete valuation field of characteristic zero and of residue characteristic \(p>0\). We denote by \(I_K\) the absolute inertia group of \(K\). We consider a proper smooth, geometrically connected curve \(X\) of genus \(g\geq 2\) and we denote by \(J\) its Jacobian variety. By the Albanese morphism with respect to a given \(K\)-rational point \(P\), the curve \(X\) can be embedded into \(J\). We put \(J_P=X(\overline K)\cap J_{\text{tors}} (\overline K)\). In the paper under review, the authors, assuming that \(J\) has ordinary semistable reduction, prove that \(I_K\) acts trivially on \(J_P\), under certain mild conditions. An interesting application of this result is given in case where \(X\) is the modular curve \(X_0(N)\), where \(N\) is a prime number \(\geq 23\) and \(P\) is one of the two cusps of \(X_0(N)\). More precisely, the authors prove, under the above conditions, that \(J_P=\)\{the two cusps\}, if \(N \notin \{23,29,31, 41,47,59,71\}\) and \(J_P=\)\{the two cusps\}\(\cup W\), otherwise, where \(W\) denotes the set of Weierstrass points of \(X_0(N)\).
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 1820779
zbMATH Keywords: torsion points
zbMATH Keywords: jacobian variety
zbMATH Keywords: Albanese morphism
zbMATH Keywords: Weierstrass points
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
cites work: Points S-entiers des courbes elliptiques. (S-integral points of elliptic curves)
cites work: Ramified torsion points on curves
cites work: Q4263347
cites work: The cuspidal torsion packet on the Fermat curve
cites work: Q5340151
cites work: The irreducibility of the space of curves of a given genus
cites work: Q2552492
cites work: Trigonal modular curves
cites work: Q3952229
cites work: Another proof of the existence of special divisors
cites work: Modular curves and the Eisenstein ideal
cites work: Rational isogenies of prime degree. (With an appendix by D. Goldfeld)
cites work: Arithmetic of Weil curves
cites work: Hyperelliptic modular curves
cites work: Spécialisation du foncteur de Picard
cites work: Courbes sur une variété abélienne et points de torsion
cites work: On modular representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) arising from modular forms
cites work: Q4702589
cites work: On the modular representations of degree two of \(\text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\)
cites work: Q3710637
cites work: Q4317834
cites work: Group schemes of prime order
description: scientific article; zbMATH DE number 1820779