Romyar T. Sharifi

Massey products and ideal class groups

Abstract We consider certain Massey products in the cohomology of a Galois extension of fields with coefficients in p-power roots of unity. We prove formulas for these products both in general and in the special case that the Galois extension in question is the maximal extension of a number field unramified outside a set of primes S including those above p and any archimedean places. We then consider those ℤ p -Kummer extensions L ∞ of the maximal p-cyclotomic extension K ∞ of a number field K that are unramified outside S. We show that Massey products describe the structure of a certain “decomposition-free” quotient of a graded piece of the maximal unramified abelian pro-p extension of L ∞ in which all primes above those in S split completely, with the grading arising from the augmentation filtration on the group ring of the Galois group of L ∞/K ∞. We explicitly describe examples of the maximal unramified abelian pro-p extensions of unramified outside p Kummer extensions of the cyclotomic field of all p-power roots of unity, for irregular primes p.

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 Kidas formula for the growth of lambda-invariants in cyclotomic Z_p-extensions of CM-fields. In fact, we give a new proof of Kidas 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.

A cup product in the Galois cohomology of number fields

This paper is devoted to the consideration of a certain cup product in the Galois cohomology of an algebraic number field with restricted ramification. Let n be a positive integer, let K be a number field containing the group μn of nth roots of unity, and let S be a finite set of primes including those above n and all real archimedean places. Let GK,S denote the Galois group of the maximal extension of K unramified outside S (inside a fixed algebraic closure of K). We consider the cup product H(GK,S, μn)⊗H(GK,S, μn)→ H(GK,S, μ⊗2 n ). ()

Iwasawa theory and the Eisenstein ideal

We verify, for each odd prime p < 1000, a conjecture of W. McCallum and the author’s 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 Q(μp) with μp-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 Zp-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 that 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.

Modular Symbols in Iwasawa Theory

This survey paper is focused on a connection between the geometry of \(\mathop{\text{GL}}\nolimits _{d}\) and the arithmetic of \(\mathop{\text{GL}}\nolimits _{d-1}\) over global fields, for integers d ≥ 2. For d = 2 over \(\mathbb{Q}\), there is an explicit conjecture of the third author relating the geometry of modular curves and the arithmetic of cyclotomic fields, and it is proven in many instances by the work of the first two authors. The paper is divided into three parts: in the first, we explain the conjecture of the third author and the main result of the first two authors on it. In the second, we explain an analogous conjecture and result for d = 2 over \(\mathbb{F}_{q}(t)\). In the third, we pose questions for general d over the rationals, imaginary quadratic fields, and global function fields.