A simple proof of a lemma of Coleman (Q2710553)

This pretty paper offers a fresh look at the local units in the \(p\)-cyclotomic tower over \(\mathbb Q\) in the spirit of Coates, Wiles, and Coleman. It gives a simpler and more explicit proof of a central technical result due to Coleman, by taking up certain explicit local units \(\alpha_n =\beta-v_n\), found by Coates and Wiles twenty-five years ago, that align into a norm-coherent system \(\alpha\) and do not come from cyclotomic units. More precisely, let \(p\) be an odd prime, \(U_n\) the group of local units in the field \(\mathbb Q\)\((\zeta_{p^{n+1}})\), and \(U_\infty\) the projective limit of the \(U_n\). It is easily shown that every \(u\in U_\infty\) has at most one Coleman series \(f_u\), and the above mentioned explicit units \(\alpha_n\) (which come about as torsion points of a suitable Lubin-Tate group law) permit to see that indeed every \(u\) does have a Coleman series. One has to admit that for this, work of Coates and Wiles is quoted, so the paper under review is not totally self-contained. The important result of Coleman alluded to above, and reproved in the paper under review, now describes the kernel and cokernel of the map \(l_\infty\), where \(l_\infty\) maps \(u\) to \(\log f_u(T) - p^{-1}\log f_u((1+T)^p-1)\). (Outside the Teichmüller character, \(l_\infty\) is an isomorphism.) The difficult point is to show that the image of \(l_\infty\) is large, and this is achieved by using that certain logarithmic derivatives of \(\alpha\) are \(p\)-adic units. In conclusion it should be recalled that Coleman's result is more general since it covers division fields of arbitrary Lubin-Tate groups, but the case at hand (where the Lubin-Tate group is the multiplicative group, and the torsion points are just roots of unity) is certainly of foremost importance in Iwasawa theory.