Iwasawa theory and the Eisenstein ideal

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.Comment: 37 page

A reciprocity map and the two variable p-adic L-function

For primes p greater than 3, we propose a conjecture that relates the values of cup products in the Galois cohomology of the maximal unramified outside p extension of a cyclotomic field on cyclotomic p-units to the values of p-adic L-functions of cuspidal eigenforms that satisfy mod p congruences with Eisenstein series. Passing up the cyclotomic and Hida towers, we construct an isomorphism of certain spaces that allows us to compare the value of a reciprocity map on a particular norm compatible system of p-units to what is essentially the two-variable p-adic L-function of Mazur and Kitagawa.Comment: 55 page

On the failure of pseudo-nullity of Iwasawa modules

We consider the family of CM-fields which are pro-p p-adic Lie extensions of number fields of dimension at least two, which contain the cyclotomic Z_p-extension, and which are ramified at only finitely many primes. We show that the Galois groups of the maximal unramified abelian pro-p extensions of these fields are not always pseudo-null as Iwasawa modules for the Iwasawa algebras of the given p-adic Lie groups. The proof uses Kida's formula for the growth of lambda-invariants in cyclotomic Z_p-extensions of CM-fields. In fact, we give a new proof of Kida's formula which includes a slight weakening of the usual assumption that mu is trivial. This proof uses certain exact sequences involving Iwasawa modules in procyclic extensions. These sequences are derived in an appendix by the second author.Comment: 26 page

A Cup Product in the Galois Cohomology of Number Fields

Let K be a number field containing the group of n-th roots of unity and S a set of primes of K including all those dividing n and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal S-ramified extension of K with coefficients in n-th roots of unity, which yields a pairing on a subgroup of the multiplicative group of K containing the S-units. In this general situation, we determine a formula for the cup product of two elements which pair trivially at all local places. Our primary focus is the case that K is the cyclotomic field of p-th roots of unity for n = p an odd prime and S consists of the unique prime above p in K. We describe a formula for this cup product in the case that one element is a p-th root of unity. We explain a conjectural calculation of the restriction of the cup product to p-units for all p < 10,000 and conjecture its surjectivity for all p satisfying Vandiver's conjecture. We prove this for the smallest irregular prime p = 37, via a computation related to the Galois module structure of p-units in the unramified extension of K of degree p. We describe a number of applications: to a product map in K-theory, to the structure of S-class groups in Kummer extensions of K, to relations in the Galois group of the maximal pro-p extension of K unramified outside p, to relations in the graded Z_p-Lie algebra associated to the representation of the absolute Galois group of Q in the outer automorphism group of the pro-p fundamental group of P^1 minus three points, and to Greenberg's pseudo-nullity conjecture.Comment: final versio