Galois theory and torsion points on curves

Matthew H. Baker; Kenneth A. Ribet

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 1, page 11-32
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Baker, Matthew H., and Ribet, Kenneth A.. "Galois theory and torsion points on curves." Journal de théorie des nombres de Bordeaux 15.1 (2003): 11-32. <http://eudml.org/doc/249096>.

@article{Baker2003,
abstract = {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.},
author = {Baker, Matthew H., Ribet, Kenneth A.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {modular curve; torsion point; abelian variety; hyperelliptic curve; cuspidal subgroup},
language = {eng},
number = {1},
pages = {11-32},
publisher = {Université Bordeaux I},
title = {Galois theory and torsion points on curves},
url = {http://eudml.org/doc/249096},
volume = {15},
year = {2003},
}

TY - JOUR
AU - Baker, Matthew H.
AU - Ribet, Kenneth A.
TI - Galois theory and torsion points on curves
JO - Journal de théorie des nombres de Bordeaux
PY - 2003
PB - Université Bordeaux I
VL - 15
IS - 1
SP - 11
EP - 32
AB - 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.
LA - eng
KW - modular curve; torsion point; abelian variety; hyperelliptic curve; cuspidal subgroup
UR - http://eudml.org/doc/249096
ER -

References

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