CHINBURG 3RD INVARIANT IN THE FACTORIZABILITY DEFECT CLASS GROUP

Chinburg's third invariant OMEGA(N/K,3) is-an-element-of Cl(Z[GAMMA]) of a Galois extension N/K of number fields with group GAMMA 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 OMEGA(N/K, 3) modulo D(ZGAMMA) 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 OMEGA(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.