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Ordinary \(p\)-adic étale cohomology groups attached to towers of elliptic modular curves. II. (English) Zbl 0967.11016
In two previous papers [J. Reine Angew. Math. 463, 49-98 (1995; Zbl 0827.11025)] and [Comp. Math. 115, 241-301 (1999; Zbl 0967.11015)] the author studied the \(p\)-adic Hodge structure of the ordinary part of the (generalized) \(p\)-adic Eichler-Shimura cohomology groups. In those papers the \(\omega ^i\)-eigenspaces for the action of \({\mathbb F}_p^{\times}\) with \(i\equiv 0, -1 \pmod{p-1}, \omega :{\mathbb F}_p^{\times}\rightarrow {\mathbb Z}_p^{\times}\) the Teichmüller character, were excluded. In the present paper that restriction is removed. Whereas in the previous work certain ‘good quotients’ of (generalized) Jacobians of modular curves were employed, now also quotients which have bad reduction at \(p\) enter the picture. The result is applied in the construction of large abelian \(p\)-extensions over cyclotomic \({\mathbb Z}_p\)-extensions of abelian number fields.

11F33 Congruences for modular and \(p\)-adic modular forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11R23 Iwasawa theory
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