no. 3
Algebraic Geometry/Group Theory
Jacobians of modular curves associated to normalizers of Cartan subgroups of level pn
[Jacobiennes de courbes modulaires associées aux normalisateurs de sous-groupes de Cartan de niveau pn.]
Présenté par : Jean-Pierre Serre
Chen, Imin1
1 Department of Mathematics, Simon Fraser University, Burnaby V5A 1S6, B.C., Canada
Comptes Rendus. Mathématique, Tome 339 (2004) no. 3, pp. 187-192

We derive a relation between induced representations of the group GL2(Z/pnZ) which implies a relation between the Jacobians of certain modular curves of level pn. A consequence of this relation is that the Jacobian of the modular curve associated to the normalizer of a non-split Cartan subgroup of GL2(Z/pnZ) does not have any non-zero rank 0 quotient defined over Q if the Birch and Swinnerton–Dyer conjecture holds for Abelian varieties.

Nous établissons une relation entre des représentations induites du groupe GL2(Z/pnZ), ce qui implique une relation entre les jacobiennes de certaines courbes modulaires de niveau pn. Une conséquence de cette relation est que la jacobienne de la courbe modulaire associée au normalisateur d'un sous-groupe Cartan non-déployé de GL2(Z/pnZ) 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.04.027

Chen, Imin 1

1 Department of Mathematics, Simon Fraser University, Burnaby V5A 1S6, B.C., Canada 
@article{CRMATH_2004__339_3_187_0,
     author = {Chen, Imin},
     title = {Jacobians of modular curves associated to normalizers of {Cartan} subgroups of level $ {p}^{n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {187--192},
     year = {2004},
     publisher = {Elsevier},
     volume = {339},
     number = {3},
     doi = {10.1016/j.crma.2004.04.027},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.crma.2004.04.027/}
}
TY  - JOUR
AU  - Chen, Imin
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
UR  - https://www.numdam.org/articles/10.1016/j.crma.2004.04.027/
DO  - 10.1016/j.crma.2004.04.027
LA  - en
ID  - CRMATH_2004__339_3_187_0
ER  - 
%0 Journal Article
%A Chen, Imin
%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
%U https://www.numdam.org/articles/10.1016/j.crma.2004.04.027/
%R 10.1016/j.crma.2004.04.027
%G en
%F CRMATH_2004__339_3_187_0
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

[1] Atkin, A.O.L.; Lehner, J. Hecke operators on Γ0(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 p2, J. Théor. Nombres Bordeaux, accepted for publication 15 January 2003, 11 p

[5] Darmon, H. The equations xn+yn=z2 and xn+yn=z3, 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 X0(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

Cité par Sources :