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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 $$\text{GL}_2(\mathbb Z/p^2\mathbb Z)$$.
MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 14G35 Modular and Shimura varieties 14H40 Jacobians, Prym varieties
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References:
 [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
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