Companion forms and the structure of $p$-adic Hecke algebras II

Abstract

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, 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 ${\Gamma }_1(N)\bigcap {\Gamma }_0(p)$. We also describe the Iwasawa module attached to the cyclotomic $Z_p$-extension of an abelian number field in terms of the Eisenstein ideal, when an appropriate Eiesenstein component is Gorenstein.