Iwasawa theory and the Eisenstein ideal

Abstract

In this paper, we relate three objects. The first is a particular value of a cup product in the cohomology of the Galois group of the maximal unramified outside p extension of a cyclotomic field containing the pth roots of unity. The second is an Iwasawa module over a nonabelian extension of the rationals, a subquotient of the maximal pro-p abelian unramified completely split at p extension of a certain pro-p Kummer extension of a cyclotomic field that contains all p-power roots of unity. The third is the quotient of an Eisenstein ideal in an ordinary Hecke algebra of Hida by the square of the Eisenstein ideal and the element given by the pth Hecke operator minus one. For the relationship between the latter two objects, we employ the work of Ohta, in which he considered a certain Galois action on an inverse limit of cohomology groups to reestablish the Main Conjecture (for p at least 5) in the spirit of the Mazur-Wiles proof. For the relationship between the former two objects, we construct an analogue to the global reciprocity map for extensions with restricted ramification. These relationships, and a computation in the Hecke algebra, allow us to prove an earlier conjecture of McCallum and the author on the surjectivity of a pairing formed from the cup product for p < 1000. We give one other application, determining the structure of Selmer groups of the modular representation considered by Ohta modulo the Eisenstein ideal.