Jacobians of modular curves associated to normalizers of Cartan subgroups of level $ {p}^{n}$

Chen, Imin

doi:10.1016/j.crma.2004.04.027

Algebraic Geometry/Group Theory

Jacobians of modular curves associated to normalizers of Cartan subgroups of level

p^{n}

[Jacobiennes de courbes modulaires associées aux normalisateurs de sous-groupes de Cartan de niveau

p^{n}

Présenté par : Jean-Pierre Serre

Chen, Imin ¹

¹ Department of Mathematics, Simon Fraser University, Burnaby V5A 1S6, B.C., Canada

Comptes Rendus. Mathématique, Tome 339 (2004) no. 3, pp. 187-192.

Résumé
Abstract

Nous établissons une relation entre des représentations induites du groupe ${GL}_{2} (Z / p^{n} Z)$ , ce qui implique une relation entre les jacobiennes de certaines courbes modulaires de niveau $p^{n}$ . Une conséquence de cette relation est que la jacobienne de la courbe modulaire associée au normalisateur d'un sous-groupe Cartan non-déployé de ${GL}_{2} (Z / p^{n} Z)$ n'a aucun quotient non-nul de rang 0 défini sur $Q$ si l'on admet la conjecture de Birch et Swinnerton–Dyer pour les variétés abéliennes.

We derive a relation between induced representations of the group ${GL}_{2} (Z / p^{n} Z)$ which implies a relation between the Jacobians of certain modular curves of level $p^{n}$ . A consequence of this relation is that the Jacobian of the modular curve associated to the normalizer of a non-split Cartan subgroup of ${GL}_{2} (Z / p^{n} Z)$ does not have any non-zero rank 0 quotient defined over $Q$ if the Birch and Swinnerton–Dyer conjecture holds for Abelian varieties.

Reçu le : 2003-07-24
Accepté le : 2004-04-30
Publié le : 2004-07-23

DOI : 10.1016/j.crma.2004.04.027

Affiliations des auteurs :

Chen, Imin ¹

¹ Department of Mathematics, Simon Fraser University, Burnaby V5A 1S6, B.C., Canada

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     title = {Jacobians of modular curves associated to normalizers of {Cartan} subgroups of level $ {p}^{n}$},
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Chen, Imin. Jacobians of modular curves associated to normalizers of Cartan subgroups of level $ {p}^{n}$. Comptes Rendus. Mathématique, Tome 339 (2004) no. 3, pp. 187-192. doi : 10.1016/j.crma.2004.04.027. http://www.numdam.org/articles/10.1016/j.crma.2004.04.027/

Bibliographie
Cité par

[1] Atkin, A.O.L.; Lehner, J. Hecke operators on $Γ_{0} (m)$ , Math. Ann., Volume 185 (1970), pp. 134-160

[2] Chen, I. The Jacobians of non-split Cartan modular curves, Proc. London Math. Soc., Volume 77 (1998) no. 1, pp. 1-38

[3] Chen, I. 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 $p^{2}$ , J. Théor. Nombres Bordeaux, accepted for publication 15 January 2003, 11 p

[5] Darmon, H. The equations $x^{n} + y^{n} = z^{2}$ and $x^{n} + y^{n} = z^{3}$ , Int. Math. Res. Notices, Volume 72 (1993) no. 1, pp. 263-273

[6] de Smit, B.; Edixhoven, S. Sur un résultat d'Imin Chen, Math. Res. Lett., Volume 7 (2000) no. 2/3, pp. 147-153

[7] Gross, B. Arithmetic on Elliptic Curves with Complex Multiplication, Lecture Notes in Math., vol. 776, Springer-Verlag, 1980

[8] Katz, N.; Mazur, B. Arithmetic Moduli of Elliptic Curves, Ann. of Math. Stud., vol. 108, Princeton University Press, 1985

[9] Mazur, B. Modular curves and the Eisenstein ideal, Publ. Math. I.H.E.S., Volume 47 (1977), pp. 33-186

[10] Mazur, B. Rational isogenies of prime degree, Invent. Math., Volume 44 (1978), pp. 129-162

[11] Mérel, L. Arithmetic of elliptic curves and diophantine equations, J. Théor. Nombres Bordeaux, Volume 11 (1999) no. 1, pp. 173-200

[12] Ribet, K. Twists of modular forms and endomorphisms of abelian varieties, Math. Ann., Volume 253 (1980), pp. 43-62

[13] D. Roberts, Shimura curves analogous to $X_{0} (N)$ , PhD thesis, Harvard University, 1989

[14] Sauvageot, F. Représentation de Steinberg et identités de projecteurs, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 505-508

[15] Serre, J.-P. Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math., Volume 15 (1972), pp. 259-331

[16] Shimura, G. Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton University Press, 1971

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