ON UNIVERSAL NORM ELEMENTS AND A PROBLEM OF COLEMAN

Suppose that n>0 kn is the cyclotomic Zp-extension of a number field k. In 1985, R. Coleman asked whether the quotient of the group (n0^Nk_n/kkn^×) ∩ U_k (the group of units of k lying in N_kn/kkn^× for all n, where N_kn/k is the norm mapping and k_n is an intermediate field) over the group of universal norm units ∩ n0^Nk_n/kU_n, where U_n is the unit group of k_n, is finite. We discuss Coleman’s problem for both the global units and the p-units, using an interpretation of the Kuz’min–Gross conjecture. Coleman claims that the quotient is finite modulo Leopoldt’s conjecture and Kuz’min–Gross’ conjecture under a mild condition. In this paper we improve Coleman’s claim by proving the claim modulo only Kuz’min–Gross’ conjecture without Leopoldt’s conjecture under the same mild condition.