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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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##### References:
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