Some new maps and ideals in classical Iwasawa theory with applications
Volume 162 / 2014
Acta Arithmetica 162 (2014), 101-140
MSC: Primary 11R18, 11R23; Secondary 11R27, 11R29, 11R42.
DOI: 10.4064/aa162-2-1
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 $ {\mathbb {Z}}_p$-tower of abelian fields, and $ {\mathfrak {J}}$ is linked by explicit reciprocity to a third ideal $ {\mathfrak {S}}$ studied more generally in [D. Solomon, Acta Arith. 143 (2010)]. 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 $\varLambda $-torsion submodules, and a new exact sequence related to the Main Conjecture.
