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Familles de polynomes liees aux courbes modulaires X_1(l) unicursales et points rationnels non-triviaux de courbes elliptiques quotient

Franck Leprevost·Michael E. Pohst·Andreas M. Schoepp
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Abstract

Soit l un entier et E_{c,l} la famille de Kubert des courbes elliptiques definies sur Q munies d'un point rationnel A d'ordre l. On note F_{c,l} la courbe elliptique quotient de E_{c,l} par le groupe engendre' par A, et f_l l'isogenie de E_{c,l} sur F_{c,l}. Pour l = 3, 4, 5 et 6, nous construisons explicitement, pour des parametrisations convenables de c, des elements non-triviaux de F_{c,l}(Q) / (f_l(E_{c,l}(Q))), autrement dit, des points explicites de F_{c,l}(Q) qui ne sont l'image par f_l d'aucun element de E_{c,l}(Q). Ces points sont en general d'ordre infini. Nous donnons des applications de cette methode a la construction d'extensions cycliques de Q de degre' l, et retrouvons certains corps obtenus par Shanks et Gras. Dans un article ulterieur, nous etudierons les proprietes arithmetiques de certaines des extensions obtenues ici.

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BibTeXRIS

Franck Leprevost, Michael E. Pohst, Andreas M. Schoepp. 2003-01-31. Familles de polynomes liees aux courbes modulaires X_1(l) unicursales et points rationnels non-triviaux de courbes elliptiques quotient. https://doi.org/10.4064/aa110-4-6

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