Some new maps and ideals in classical Iwasawa theory with applications

David Solomon¹

¹ 132 Hillfield Ave. London N8 7DJ, United Kingdom

Acta Arithmetica, Tome 162 (2014) no. 2, pp. 101-140.

Résumé

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.

Détail

DOI : 10.4064/aa162-2-1

Keywords: introduce ideal mathfrak p adic galois group ring associated real abelian field related ideal mathfrak imaginary abelian fields result equivariant kummer type pairing applied stark units mathbb p tower abelian fields mathfrak linked explicit reciprocity third ideal mathfrak studied generally solomon acta arith leads unifying framework iwasawa theory fields including real analogue stickelbergers theorem links certain fitting ideals varlambda torsion submodules exact sequence related main conjecture