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Stark units and the main conjectures for totally real fields

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  23 October 2009

Kâzım Büyükboduk
Kâzım Büyükboduk*
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA (email: kazim@math.stanford.edu)
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Abstract

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The main theorem of the author’s thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed by the author in an earlier paper, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this is indeed the case. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units provides the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement in the context of local Iwasawa theory, assuming the truth of the Rubin–Stark conjecture and Leopoldt’s conjecture. This statement in the local Iwasawa theory context turns out to be interesting in its own right, as it suggests a relation between the solutions to p-adic and complex Stark conjectures.

Keywords

Iwasawa theoryKolyvagin systemsStark conjectures

MSC classification

Secondary: 11R23: Iwasawa theory 11R27: Units and factorization 11R29: Class numbers, class groups, discriminants 11R42: Zeta functions and $L$-functions of number fields
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 5 , September 2009 , pp. 1163 - 1195
Copyright
Copyright © Foundation Compositio Mathematica 2009

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