Chinburg's Third Invariant in the Factorisability Defect Class Group

Holland, D.

Canadian journal of mathematics, Tome 46 (1994) no. 2, pp. 324-342

Résumé

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

Détail

DOI : 10.4153/CJM-1994-016-0

Mots-clés : 11R33, 11R65, 16E30, 11R18, factorisability, defect class group, Euler systems, multiplicative Galois structure, Chinburg conjecture

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