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

Imin Chen

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

Imin Chen ¹

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

Comptes Rendus. Mathématique, Volume 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 :

Imin Chen ¹

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

@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},
}

TY  - JOUR
AU  - Imin Chen
TI  - Jacobians of modular curves associated to normalizers of Cartan subgroups of level $ {p}^{n}$
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 187
EP  - 192
VL  - 339
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2004.04.027
LA  - en
ID  - CRMATH_2004__339_3_187_0
ER  -

%0 Journal Article
%A Imin Chen
%T Jacobians of modular curves associated to normalizers of Cartan subgroups of level $ {p}^{n}$
%J Comptes Rendus. Mathématique
%D 2004
%P 187-192
%V 339
%N 3
%I Elsevier
%R 10.1016/j.crma.2004.04.027
%G en
%F CRMATH_2004__339_3_187_0

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/

Bibliographie
Cité par

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :

Commentaires - Politique