Rational points on Atkin-Lehner quotients of Shimura curves of discriminant p q

Florence Gillibert[1]

  • [1] IMB Bordeaux I 351, cours de la Libération 33405 Talence (France)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 4, page 1613-1649
  • ISSN: 0373-0956

Abstract

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Let p and q be two distinct prime numbers, and X p q / w q be the quotient of the Shimura curve of discriminant p q by the Atkin-Lehner involution w q . We describe a way to verify in wide generality a criterion of Parent and Yafaev to prove that if p and q satisfy some explicite congruence conditions, known as the conditions of the non ramified case of Ogg, and if p is large enough compared to q , then the quotient X p q / w q has no rational point, except possibly special points.

How to cite

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Gillibert, Florence. "Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant $pq$." Annales de l’institut Fourier 63.4 (2013): 1613-1649. <http://eudml.org/doc/275488>.

@article{Gillibert2013,
abstract = {Soient $p$ et $q$ deux nombres premiers distincts et $X^\{pq\}/w_q$ le quotient de la courbe de Shimura de discriminant $pq$ par l’involution d’Atkin-Lehner $w_q$. Nous décrivons un moyen permettant de vérifier un critère de Parent et Yafaev en grande généralité pour prouver que si $p$ et $q$ satisfont des conditions de congruence explicites, connues comme les conditions du cas non ramifié de Ogg, et si $p$ est assez grand par rapport à $q$, alors le quotient $X^\{pq\}/w_q$ n’a pas de point rationnel non spécial.},
affiliation = {IMB Bordeaux I 351, cours de la Libération 33405 Talence (France)},
author = {Gillibert, Florence},
journal = {Annales de l’institut Fourier},
keywords = {Shimura curves; rational points; Gross vectors; Atkin-Lehner involutions},
language = {fre},
number = {4},
pages = {1613-1649},
publisher = {Association des Annales de l’institut Fourier},
title = {Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant $pq$},
url = {http://eudml.org/doc/275488},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Gillibert, Florence
TI - Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant $pq$
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 4
SP - 1613
EP - 1649
AB - Soient $p$ et $q$ deux nombres premiers distincts et $X^{pq}/w_q$ le quotient de la courbe de Shimura de discriminant $pq$ par l’involution d’Atkin-Lehner $w_q$. Nous décrivons un moyen permettant de vérifier un critère de Parent et Yafaev en grande généralité pour prouver que si $p$ et $q$ satisfont des conditions de congruence explicites, connues comme les conditions du cas non ramifié de Ogg, et si $p$ est assez grand par rapport à $q$, alors le quotient $X^{pq}/w_q$ n’a pas de point rationnel non spécial.
LA - fre
KW - Shimura curves; rational points; Gross vectors; Atkin-Lehner involutions
UR - http://eudml.org/doc/275488
ER -

References

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