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Chinburg's Third Invariant in the Factorisability Defect Class Group

Published online by Cambridge University Press:  20 November 2018

D. Holland
D. Holland*
Affiliation:
Department of Mathematics and Statistics McMaster University 1280 Main Street West Hamilton, Ontario L8S4K1
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Abstract

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Chinburg's third invariant Ω(N/K, 3) ∊ C1(Z[Γ]) of a Galois extension N/K of number fields with group Γ is closely related to the Galois structure of unit groups and ideal class groups, and deep unsolved problems such as Stark's conjecture.

We give a formula for Ω(N/K, 3) modulo D(ZΓ) in the factorisability defect class group, reminiscent of analytic class number formulas. Specialising to the case of an absolutely abelian, real field N, we give a natural conjecture in terms of Hecke factorisations which implies the vanishing of the invariant in the defect class group.

We prove this conjecture when N has prime-power conductor using Euler systems of cyclotomic units, Ramachandra units and Hecke factorisation. This supports a general conjecture of Chinburg, which in our situation specialises to the statement that Ω(N/K, 3) = 0 for such extensions.

We also develop a slightly extended version of Euler systems of units for general abelian extensions, which will be applied to abelian extensions of imaginary quadratic fields elsewhere

Keywords

11R3311R6516E3011R18factorisabilitydefect class groupEuler systemsmultiplicative Galois structureChinburg conjecture
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 2 , 01 April 1994 , pp. 324 - 342
Copyright
Copyright © Canadian Mathematical Society 1994

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