We bound the j-invariant of integral points on a modular curve in terms of the congruence group defining the curve. We apply this to prove that, under the GRH, the modular curve has no non-trivial rational point if p is a sufficiently large prime number.
On borne l'invariant j des points entiers des courbes modulaires, en fonction du groupe de congruence définissant la courbe. Sous l'hypothèse de Riemann généralisée, on en déduit que, si p est un nombre premier suffisamment grand, la courbe modulaire n'a pas de point rationnel non trivial.
Accepted:
Published online:
Yu. Bilu 1; Pierre Parent 1
@article{CRMATH_2008__346_11-12_599_0, author = {Yu. Bilu and Pierre Parent}, title = {Integral \protect\emph{j}-invariants and {Cartan} structures for elliptic curves}, journal = {Comptes Rendus. Math\'ematique}, pages = {599--602}, publisher = {Elsevier}, volume = {346}, number = {11-12}, year = {2008}, doi = {10.1016/j.crma.2008.04.002}, language = {en}, }
Yu. Bilu; Pierre Parent. Integral j-invariants and Cartan structures for elliptic curves. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 599-602. doi : 10.1016/j.crma.2008.04.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.04.002/
[1] Yu. Bilu, P. Parent, Explicit bounds for integral j-invariants and level of Cartan structures for elliptic curves, in preparation
[2] On Weil's “théorème de décomposition”, Amer. J. Math., Volume 105 (1983), pp. 295-308
[3] Sur les modules de torsion des courbes elliptiques, Math. Ann., Volume 310 (1998), pp. 47-54
[4] Modular Units, Grundlehren der Mathematischen Wissenschaften, vol. 244, Springer-Verlag, New York-Berlin, 1981 (xiii+358 pp)
[5] Galois properties of division fields of elliptic curves, Bull. London Math. Soc., Volume 25 (1993), pp. 247-254
[6] L. Merel, Normalizers of split Cartan subgroups and supersingular elliptic curves, in: Proceedings of the conference, Diophantine Geometry, Pisa, 2005
[7] Sur une majoration explicite pour un degré d'isogénie liant deux courbes elliptiques, Acta Arith., Volume 100 (2001), pp. 203-243
[8] Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math., Volume 15 (1972), pp. 259-331
[9] Quelques applications du théorème de densité de Chebotarev, Publ. Math. IHES, Volume 54 (1981), pp. 323-401
[10] Introduction to the Arithmetic Theory of Automorphic Functions, Publ. Math. Soc. Japan, vol. 11, Iwanami Shoten, Tokyo, 1971 (Princeton University Press, Princeton, NJ)
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