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
