Iwasawa theory and the Eisenstein ideal

Abstract

We verify, for each odd prime $p \le 1000$, a conjecture of W. G. McCallum and R. T. Sharifi on the surjectivity of pairings on $p$-units constructed out of the cup product on the first Galois cohomology group of the maximal unramified outside $p$ extension of ${\bf Q}(\mu_p)$ with $\mu_p$-coefficients. In the course of the proof, we relate several Iwasawa-theoretic and Hida-theoretic objects. In particular, we construct a canonical isomorphism between an Eisenstein ideal modulo its square and the second graded piece in an augmentation filtration of a classical Iwasawa module over an abelian pro-$p$ Kummer extension of the cyclotomic ${\bf Z}_p$-extension of an abelian field. This Kummer extension arises from the Galois representation on an inverse limit of ordinary parts of first cohomology groups of modular curves which was considered by M. Ohta in order to give another proof of the Iwasawa main conjecture in the spirit of that of B. Mazur and A. Wiles. In turn, we relate the Iwasawa module over the Kummer extension to the quotient of the tensor product of the classical cyclotomic Iwasawa module and the Galois group of the Kummer extension by the image of a certain reciprocity map that is constructed out of an inverse limit of cup products up the cyclotomic tower. We give an application to the structure of the Selmer groups of Ohta's modular representation taken modulo the Eisenstein ideal