Some New Ideals in Classical Iwasawa Theory

We construct an analogue of the Stickelberger ideal for real abelian fields by means of cyclotomic units and a Kummer-type pairing with values in a completed p-adic group-ring. We give several different descriptions of this ideal, prove the analogue of Stickelberger's Theorem using Thaine's methods and establish links with certain Fitting ideals in a particular case. Our construction fits into a new Iwasawa-theoretic framework including two related ideals for imaginary abelian fields, one linked by explicit reciprocity to an ideal studied more generally in [S3]. It also has applications to Λ-torsion submodules and gives a new exact sequence related to the Main Conjecture.