Some New Maps and Ideals in Classical Iwasawa Theory with Applications

Abstract

We introduce a new ideal {\mathfrak D} of the p-adic Galois group-ring associated to a real abelian field and a related ideal {\mathfrak J} for imaginary abelian fields. Both result from an equivariant, Kummer-type pairing applied to Stark units in a Z_p-tower of abelian fields and {\mathfrak J} is linked by explicit reciprocity to a third ideal {\mathfrak S} studied more generally in a previous work. This leads to a new and unifying framework for the Iwasawa Theory of such fields including a real analogue of Stickelberger's Theorem, links with certain Fitting ideals and Λ-torsion submodules, and a new exact sequence related to the Main Conjecture.

36 pages. Several additions and improvements, principally: new title; new section on variation of K and kernel of {\mathfrak j}; more detail on links with Greenberg's Conjecture; more on Λ-torsion including more explicit determination of Λ-torsion submodule of {\mathfrak X}_\infty; various remarks added