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Some new maps and ideals in classical Iwasawa theory with applications

Volume 162 / 2014

David Solomon 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.

Authors

  • David Solomon132 Hillfield Ave.
    London N8 7DJ, United Kingdom
    e-mail

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