Galois theory and torsion points on curves

Baker, Matthew H.; Ribet, Kenneth A.

Baker, Matthew H. ; Ribet, Kenneth A.

Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 11-32.

Résumé
Abstract

Dans cet article, nous exposons diverses techniques de théorie de Galois qui s’appliquent à l’étude des points de torsion des courbes. En particulier, nous donnons de nouvelles démonstrations de résultats de A. Tamagawa et des auteurs concernant les points de torsion des courbes à “bonne” ou “semi-stable” réduction “ordinaire” en un nombre premier donné. Nous donnons également de nouvelles démonstrations de : (1) la conjecture de Manin-Mumford : il n’y a qu’un nombre fini de points de torsion sur une courbe de genre au moins $2$ plongée dans sa jacobienne par l’application d’Albanese ; et (2) : la conjecture de Coleman-Kaskel-Ribet : pour un nombre premier $p \geq 23$ , les seuls points de torsion appartenant à la courbe $X_{0} (p)$ plongée dans sa jacobienne par un plongement cuspidal sont les pointes (et les points de branchement hyperelliptique lorsque $X_{0} (p)$ est hyperelliptique et $p \neq 37$ ). Afin de rendre l’exposition aussi utile que possible, nous donnons des références pour tous les résultats sur les courbes modulaires qui interviennent dans notre discussion.

In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present authors for studying torsion points on curves with “ordinary good” or “ordinary semistable” reduction at a given prime. We also give new proofs of : (1) the Manin-Mumford conjecture : there are only finitely many torsion points lying on a curve of genus at least $2$ embedded in its jacobian by an Albanese map; and (2) the Coleman-Kaskel-Ribet conjecture : if $p$ is a prime number which is at least $23$ , then the only torsion points lying on the curve $X_{0} (p)$ , embedded in its jacobian by a cuspidal embedding, are the cusps (together with the hyperelliptic branch points when $X_{0} (p)$ is hyperelliptic and $p$ is not 37). In an effort to make the exposition as useful as possible, we provide references for all of the facts about modular curves which are needed for our discussion.

MR Zbl | 3 citations dans Numdam

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     title = {Galois theory and torsion points on curves},
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     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {1},
     year = {2003},
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     zbl = {1065.11045},
     language = {en},
     url = {https://www.numdam.org/item/JTNB_2003__15_1_11_0/}
}

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Baker, Matthew H.; Ribet, Kenneth A. Galois theory and torsion points on curves. Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 11-32. https://www.numdam.org/item/JTNB_2003__15_1_11_0/

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