Modular forms and the number theory of cyclotomic fields.

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It is known that, if the Eisenstein components of p-adic (or A-adic) Hecke algebras attached to cusp forms are Gorenstein rings, then one can completely determine the structure of the Iwasawa modules obtained from ideal class groups of cyclotomic fields. However, this seems to be a very difficult problem. In contrast to this, the Gorenstein property of similar Hecke algebras attached to modular forms seems relatively easier to achieve. In fact, Skinner and Wiles have established such a property under certain numerical conditions. We investigated the latter problem from a point of view completely different from Skinner and Wiles, and also obtained applications to the theory of cyclotomic fields.First, as for the Gorenstein property of Hecke algebras, in connection with the companion forms in the spaces of modular forms (mod p), we proved that:・If the dimension of certain space of companion forms is one, then certain Eisenstein component of the Hecke algebra attached to modular forms is Gorenstein.As an application of our previous result, we also proved that:・If certain generalized Bernoulli number (multiplied by an elementary factor) is a p-adic unit, then the above one-dimensionality follows.These two results imply a numerical criterion as in the work, of Skinner and Wiles; but our method covers a wider class of Eisenstein components than theirs.As for the application to the theory of cyclotomic fields, we have shown, that:・Under the assumption that the Eisenstein components of the Hecke algebras attached modular forms are Gorenstein, one can explicitly describe the Iwasawa modules in terms of A-adic Hecke algebras.Our article on these results is accepted for publication in J. reine angew. Math.