We introduce the Coleman conjecture on circular distributions made around 1980's and its implications in Euler systems over Q. In our joint work with Burns, Bullach and Daoud, we will provide a proof of the Coleman conjecture and introduce "Coleman's conjecture for higher ranks" with its application to the equivariant Tamagawa Number conjecture for Dirichlet L-functions at s=0 (eTNC (G_m), in short) of Bloch, Kato, Burns and Flach.

Based on the method used in the proof of Coleman's conjecture, we provide a proof of the minus part of eTNC(G_m) for CM extensions of totally real fields unconditionally and imaginary quadratic fields under mu vanishing condition, which accordingly recovers or implies refinements and variants of ‘Stark-type conjectures’ (in particular, conjectures of Rubin-Stark, Brumer, Chinberg, and the lifted root number conjecture of Gruenberg, Rittel and Weiss, among others).