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A certain \(p\)-divisible subgroup of the Jacobian of the curve \(X_1(Np^r)\) as a module over the Hecke algebra. (Un sous-groupe \(p\)-divisible de la Jacobienne de \(X_ 1(Np^ r)\) comme module sur l’algèbre de Hecke.) (French) Zbl 0677.14006
Translation from the Russian: The modular curve \(X_r=X_1(Np^r)\), its Jacobian \(J_r\) and the \(p\)-divisible part \(J_{r,p}=J_r[p^{\infty}]\) of the Jacobian are considered. Then \(J_{r,p}\) is a module over the Hecke algebra \(h_r\) of the curve \(X_r\). The limits taken with respect to the natural coverings \(X_r\to X_s\) (for \(r\geq s\geq 1)\) \(J_{\infty,p}=\varinjlim_ rJ_{r,p}\), \(h_{\infty}=\varprojlim_r h_r\), are considered. The structure of \(J_{\infty,r}\) as a \(h_{\infty}\)-module is considered. More exactly, constructions of H. Hida [Invent. Math. 85, 545–613 (1986; Zbl 0612.10021)] are generalized and, for every local factor \(R\) of the ordinary part of the Hecke algebra, the structure of the \(R\)-module \(J_{\infty,p}(R)\)
\[ J_{\infty,p}(R) \cong R\otimes_{\Lambda}\operatorname{Hom}(\Lambda,\Pi_p) \oplus\operatorname{Hom}(R,\Pi_p), \]
where \(\Pi_p=\mathbb Q_P/\mathbb Z_P\), \(\Lambda =\mathbb Z_p[[\Gamma ]]\) is the Iwasawa algebra, is completely described.

MSC:
14H40 Jacobians, Prym varieties
11F11 Holomorphic modular forms of integral weight
14H25 Arithmetic ground fields for curves
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