Relations between jacobians of modular curves of level $p^2$

Chen, Imin; De Smit, Bart; Grabitz, Martin

doi:10.5802/jtnb.435

Relations between jacobians of modular curves of level

p^{2}

Chen, Imin ¹ ; De Smit, Bart ² ; Grabitz, Martin ³

¹ Department of Mathematics Simon Fraser University Burnaby, B.C., Canada, V5A 1S6
² Mathematisch Instituut Universiteit Leiden Postbus 9512 2300 RA Leiden, Netherlands
³ Mathematisches Institut der Humboldt Universitaet Rudower Chaussee 25 (Ecke Magnusstrasse) 12489 Berlin House 1, Germany

Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 95-106.

Résumé
Abstract

Nous établissons une relation entre les représentations induites sur le groupe ${GL}_{2} (ℤ / p^{2} ℤ)$ qui implique une relation entre les jacobiennes des certaines courbes modulaires de niveaux $p^{2}$ . La motivation de la construction de cette relation est la détermination de l’applicabilité de la méthode de Mazur à la courbe modulaire associée au normalisateur d’un subgroupe Cartan non-déployé de ${GL}_{2} (ℤ / p^{2} ℤ)$ .

We derive a relation between induced representations on the group ${GL}_{2} (ℤ / p^{2} ℤ)$ which implies a relation between the jacobians of certain modular curves of level $p^{2}$ . The motivation for the construction of this relation is the determination of the applicability of Mazur’s method to the modular curve associated to the normalizer of a non-split Cartan subgroup of ${GL}_{2} (ℤ / p^{2} ℤ)$ .

MR Zbl

DOI : 10.5802/jtnb.435

Affiliations des auteurs :

Chen, Imin ¹ ; De Smit, Bart ² ; Grabitz, Martin ³

@article{JTNB_2004__16_1_95_0,
     author = {Chen, Imin and De Smit, Bart and Grabitz, Martin},
     title = {Relations between jacobians of modular curves of level $p^2$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {95--106},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {1},
     year = {2004},
     doi = {10.5802/jtnb.435},
     zbl = {02184633},
     mrnumber = {2145574},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.435/}
}

TY  - JOUR
AU  - Chen, Imin
AU  - De Smit, Bart
AU  - Grabitz, Martin
TI  - Relations between jacobians of modular curves of level $p^2$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2004
SP  - 95
EP  - 106
VL  - 16
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.435/
DO  - 10.5802/jtnb.435
LA  - en
ID  - JTNB_2004__16_1_95_0
ER  -

%0 Journal Article
%A Chen, Imin
%A De Smit, Bart
%A Grabitz, Martin
%T Relations between jacobians of modular curves of level $p^2$
%J Journal de théorie des nombres de Bordeaux
%D 2004
%P 95-106
%V 16
%N 1
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.435/
%R 10.5802/jtnb.435
%G en
%F JTNB_2004__16_1_95_0

Chen, Imin; De Smit, Bart; Grabitz, Martin. Relations between jacobians of modular curves of level $p^2$. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 95-106. doi : 10.5802/jtnb.435. http://www.numdam.org/articles/10.5802/jtnb.435/

Bibliographie
Cité par

[1] I. Chen, Jacobians of a certain class of modular curves of level $p^{n}$ . Provisionally accepted by the Comptes Rendus de l’Academie des Sciences - Mathematics, 30 January 2004.

[2] B. de Smit and S. Edixhoven, Sur un résultat d’Imin Chen. Math. Res. Lett. 7 (2–3) (2000), 147–153. | MR | Zbl

[3] N. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves. Annals of Mathematics Studies 108. Princeton University Press, 1985. | MR | Zbl

[4] B. Mazur, Rational isogenies of prime degree. Inventiones mathematicae 44 (1978), 129–162. | MR | Zbl

[5] D. Mumford, Abelian varieties. Tata Institute of Fundamental Research Studies in Mathemaitcs 5. Oxford University Press, London, 1970. | MR | Zbl

[6] J.P. Murre, On contravariant functors from the category of preschemes over a field into the category of abelian groups. Publ. Math. IHES 23 (1964), 5–43. | Numdam | MR | Zbl

[7] F. Oort, Sur le schéma de Picard. Bull. Soc. Math. Fr. 90 (1962), 1–14. | Numdam | MR | Zbl

[8] J.P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Inventiones Mathematicae 15 (1972), 259–331. | MR | Zbl

Cité par Sources :