zbMATH — the first resource for mathematics

Relations between Jacobians of modular curves of level \(p^2\). (English) Zbl 1156.11323
Summary: We derive a relation between induced representations on the group \(\text{GL}_2(\mathbb Z/p^2\mathbb Z)\) 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 \(\text{GL}_2(\mathbb Z/p^2\mathbb Z)\).
11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
14H40 Jacobians, Prym varieties
Full Text: DOI Numdam EuDML
[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. · Zbl 0968.14024
[3] N. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves. Annals of Mathematics Studies 108. Princeton University Press, 1985. · Zbl 0576.14026
[4] B. Mazur, Rational isogenies of prime degree. Inventiones mathematicae 44 (1978), 129-162. · Zbl 0386.14009
[5] D. Mumford, Abelian varieties. Tata Institute of Fundamental Research Studies in Mathemaitcs 5. Oxford University Press, London, 1970. · Zbl 0223.14022
[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. · Zbl 0142.18402
[7] F. Oort, Sur le schéma de Picard. Bull. Soc. Math. Fr. 90 (1962), 1-14. · Zbl 0123.13901
[8] J.P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Inventiones Mathematicae 15 (1972), 259-331. · Zbl 0235.14012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.