Companion forms and the structure of p-adic Hecke algebras

Details Companion forms and the structure of p-adic Hecke algebras Companion forms and the structure of p-adic Hecke algebras

2004-03-25

arxiv.org/abs/math/0403428v2

Mathematics

Number Theory

29pages

Scientific paper

We study the structure of the Eisenstein component of Hida's ordinary p-adic Hecke algebra attached to modular forms, in connection with the companion forms in the space of modular forms (mod p). We show that such an algebra is a Gorenstein ring if certain space of modular forms (mod p) having companions is one-dimensional; and also give a numerical criterion for this one-dimensionality. This in part overlaps with a work of Skinner and Wiles; but our method, based on a work of Ulmer, is totally different. We then consider consequences of the above mentioned Gorenstein property. We especially discuss the connection with the Iwasawa theory.

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