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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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