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Iwasawa modules attached to congruences of cusp forms. (English) Zbl 0607.10022
The author studies congruences modulo \(p\) \((p\geq 5)\) of systems of cusp forms of fixed level and varying weights. To be more precise, fix an embedding of \(\bar {\mathbb Q}\) into the \(p\)-adic completion \(\Omega\) of \(\bar{\mathbb Q}_ p\), and let \(T(p)\) be the usual Hecke operator. A non-zero common eigenform \(f\) is called ordinary if its eigenvalue for \(T(p)\) is a \(p\)-adic unit in \(\Omega\). It is easy to see that the space of \(p\)-adic modular forms, and the Hecke algebra may both be viewed as modules over the one variable Iwasawa algebra \(\Lambda\). Using Katz’s theory of \(p\)-adic modular forms the author proves that the Hecke algebra \(h\) for the space of all ordinary modular forms is actually a free \(\Lambda\)-module of finite rank.
Let \(\mathcal L\) denote the quotient field of \(\Lambda\). To each simple summand \({\mathcal K}\) in \(h\otimes_{\Lambda} \mathcal L\) the author associates a finite \(p\)-power torsion \(\Lambda\)-module \(C(\mathcal K)\), the module of congruence of \(\mathcal K\). When \({\mathcal K}\) is actually a summand of the Hecke algebra (extended to \(\mathcal L\)) associated to ordinary cusp forms, the module \(C(\mathcal K)\) is shown to regulate the non-trivial congruences (modulo the maximal ideal of the integer ring of \(\Omega\)) occurring at each weight between the associated cusp forms.
Reviewer: S.Kamienny

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11F11 Holomorphic modular forms of integral weight
11R23 Iwasawa theory
11G18 Arithmetic aspects of modular and Shimura varieties
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