We study the rational torsion subgroup of the modular Jacobian variety J 0 (N ) when N is squarefree. We prove that the p-primary part of this group coincides with that of the cuspidal divisor class group for p ≥ 3 when 3 N , and for p ≥ 5 when 3 | N . We further determine the structure of each eigenspace of such p-primary part with respect to the Atkin-Lehner involutions. This is based on our study of the Eisenstein ideals in the Hecke algebras.
The purpose of this paper is to study the structure of congruence modules (or modules of congruences) associated with Eisenstein series in various contexts in the Λ-adic theory of elliptic modular forms. Under some assumptions, we explicitly describe such modules in terms of Kubota-Leopoldt p-adic L-functions. 2003 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Le but de cet article est d'étudier la structure des modules de congruence qui, dans la théorie Λ-adique des formes modulaires elliptiques, sont, de diverses manières, associées aux séries d'Eisenstein. Sous certaines conditions, nous décrivons explicitement de tels modules en termes des fonctions L p-adiques de Kubota-Leopoldt.
We study the structure of the Eiesenstein component of Hida's ordinary p-adic Hecke algebra attached to modular forms, in connection with the companion forms in the space of modular forms (mod p). We show that such an algebra is a Gorenstein ring if certain space of modular forms (mod p) having companions is one-dimensional; and also give a numerical criterion for this one-dimensionality. This in part overlaps with a work of Skinner and Wiles; but our method, based on a work of Ulmer, is totally different. We then consider consequences of the above mentioned Gorenstein property. We especially discuss the connection with the Iwasawa theory.
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