We derive a relation between induced representations of the group which implies a relation between the Jacobians of certain modular curves of level . A consequence of this relation is that the Jacobian of the modular curve associated to the normalizer of a non-split Cartan subgroup of does not have any non-zero rank 0 quotient defined over if the Birch and Swinnerton–Dyer conjecture holds for Abelian varieties.
Nous établissons une relation entre des représentations induites du groupe , ce qui implique une relation entre les jacobiennes de certaines courbes modulaires de niveau . Une conséquence de cette relation est que la jacobienne de la courbe modulaire associée au normalisateur d'un sous-groupe Cartan non-déployé de n'a aucun quotient non-nul de rang 0 défini sur si l'on admet la conjecture de Birch et Swinnerton–Dyer pour les variétés abéliennes.
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Imin Chen 1
@article{CRMATH_2004__339_3_187_0, author = {Imin Chen}, title = {Jacobians of modular curves associated to normalizers of {Cartan} subgroups of level $ {p}^{n}$}, journal = {Comptes Rendus. Math\'ematique}, pages = {187--192}, publisher = {Elsevier}, volume = {339}, number = {3}, year = {2004}, doi = {10.1016/j.crma.2004.04.027}, language = {en}, }
Imin Chen. Jacobians of modular curves associated to normalizers of Cartan subgroups of level $ {p}^{n}$. Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 187-192. doi : 10.1016/j.crma.2004.04.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.04.027/
[1] Hecke operators on , Math. Ann., Volume 185 (1970), pp. 134-160
[2] The Jacobians of non-split Cartan modular curves, Proc. London Math. Soc., Volume 77 (1998) no. 1, pp. 1-38
[3] On relations between Jacobians of certain modular curves, J. Algebra, Volume 231 (2000), pp. 414-448
[4] I. Chen, B. de Smit, M. Grabitz, Relations between Jacobians of modular curves of level , J. Théor. Nombres Bordeaux, accepted for publication 15 January 2003, 11 p
[5] The equations and , Int. Math. Res. Notices, Volume 72 (1993) no. 1, pp. 263-273
[6] Sur un résultat d'Imin Chen, Math. Res. Lett., Volume 7 (2000) no. 2/3, pp. 147-153
[7] Arithmetic on Elliptic Curves with Complex Multiplication, Lecture Notes in Math., vol. 776, Springer-Verlag, 1980
[8] Arithmetic Moduli of Elliptic Curves, Ann. of Math. Stud., vol. 108, Princeton University Press, 1985
[9] Modular curves and the Eisenstein ideal, Publ. Math. I.H.E.S., Volume 47 (1977), pp. 33-186
[10] Rational isogenies of prime degree, Invent. Math., Volume 44 (1978), pp. 129-162
[11] Arithmetic of elliptic curves and diophantine equations, J. Théor. Nombres Bordeaux, Volume 11 (1999) no. 1, pp. 173-200
[12] Twists of modular forms and endomorphisms of abelian varieties, Math. Ann., Volume 253 (1980), pp. 43-62
[13] D. Roberts, Shimura curves analogous to , PhD thesis, Harvard University, 1989
[14] Représentation de Steinberg et identités de projecteurs, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 505-508
[15] Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math., Volume 15 (1972), pp. 259-331
[16] Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton University Press, 1971
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