[Rational points on Atkin-Lehner quotients of Shimura curves of discriminant ]
Let and be two distinct prime numbers, and be the quotient of the Shimura curve of discriminant by the Atkin-Lehner involution . We describe a way to verify in wide generality a criterion of Parent and Yafaev to prove that if and satisfy some explicite congruence conditions, known as the conditions of the non ramified case of Ogg, and if is large enough compared to , then the quotient has no rational point, except possibly special points.
Soient et deux nombres premiers distincts et le quotient de la courbe de Shimura de discriminant par l’involution d’Atkin-Lehner . Nous décrivons un moyen permettant de vérifier un critère de Parent et Yafaev en grande généralité pour prouver que si et satisfont des conditions de congruence explicites, connues comme les conditions du cas non ramifié de Ogg, et si est assez grand par rapport à , alors le quotient n’a pas de point rationnel non spécial.
Mot clés : courbes de Shimura, points rationnels, vecteurs de Gross, involutions d’Atkin-Lehner
Keywords: Shimura curves, rational points, Gross vectors, Atkin-Lehner involutions
Gillibert, Florence 1
@article{AIF_2013__63_4_1613_0, author = {Gillibert, Florence}, title = {Points rationnels sur les quotients {d{\textquoteright}Atkin-Lehner} de courbes de {Shimura} de discriminant $pq$}, journal = {Annales de l'Institut Fourier}, pages = {1613--1649}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {63}, number = {4}, year = {2013}, doi = {10.5802/aif.2810}, mrnumber = {3137362}, zbl = {06359596}, language = {fr}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2810/} }
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 SP - 1613 EP - 1649 VL - 63 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2810/ DO - 10.5802/aif.2810 LA - fr ID - AIF_2013__63_4_1613_0 ER -
%0 Journal Article %A Gillibert, Florence %T Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant $pq$ %J Annales de l'Institut Fourier %D 2013 %P 1613-1649 %V 63 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2810/ %R 10.5802/aif.2810 %G fr %F AIF_2013__63_4_1613_0
Gillibert, Florence. Points rationnels sur les quotients d’Atkin-Lehner de courbes de Shimura de discriminant $pq$. Annales de l'Institut Fourier, Volume 63 (2013) no. 4, pp. 1613-1649. doi : 10.5802/aif.2810. https://aif.centre-mersenne.org/articles/10.5802/aif.2810/
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