On higher order Stickelberger-type theorems

We discuss an explicit refinement of Rubinʼs integral version of Starkʼs conjecture. We prove that this refinement is a consequence of the relevant case of the Equivariant Tamagawa Number Conjecture of Burns and Flach, hence obtaining a full proof in several important cases. We also derive several explicit consequences of this refinement concerning the annihilation as Galois modules of ideal class groups by explicit elements constructed from the values of higher order derivatives of Dirichlet L-functions. We finally describe the relation between our approach and those found in recent work of Emmons and Popescu and of Buckingham.