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**Companion forms and the structure of \(p\)-adic Hecke algebras. II.**
*(English)*
Zbl 1187.11014

Summary: The subject of this paper is to study the structure of the Eisenstein component of Hida’s universal ordinary \(p\)-adic Hecke algebra attached to modular forms (rather than cusp forms). We give a sufficient condition for such a ring to be Gorenstein in terms of companion forms in characteristic p; and also a numerical criterion which assures the validity of that condition. This type of result was already obtained in our previous work [J. Reine Angew. Math. 585, 141–172 (2005; Zbl 1081.11035)], in which two cases were left open. The purpose of this work is to extend our method to cover these remaining cases. New ingredients of the proof consist of: a new construction of a pairing between modular forms over a finite field; and a comparison result for ordinary modular forms of weight two with respect to \(\Gamma_1(N)\) and \(\Gamma(N)\cap \Gamma_0(p)\). We also describe the Iwasawa module attached to the cyclotomic \(\mathbb Z_p\)-extension of an abelian number field in terms of the Eisenstein ideal, when an appropriate Eisenstein component is Gorenstein.

### MSC:

11F33 | Congruences for modular and \(p\)-adic modular forms |

11F80 | Galois representations |

11R23 | Iwasawa theory |

11F85 | \(p\)-adic theory, local fields |