# Moscow Mathematical Journal

Volume 22, Issue 1, January–March 2022  pp. 121–132.

On Universal Norm Elements and a Problem of Coleman

Authors:  Soogil Seo (1)
Author institution:(1) Department of Mathematics, Yonsei University, 134 Sinchon-Dong, Seodaemun-Gu, Seoul 120-749, South Korea

Summary:

Suppose that $\bigcup_{n \ge 0} k_n$ is the cyclotomic $\mathbb{Z}_p$-extension of a number field $k$. In 1985, R. Coleman asked whether the quotient of the group $( \bigcap_{n\ge 0} N_{k_n/k} k_n^\times) \cap U_k$ (the group of units of $k$ lying in $N_{k_n/k} k_n^\times$ for all $n$, where $N_{k_n/k}$ is the norm mapping and $k_n$ is an intermediate field) over the group of universal norm units $\bigcap_{n\ge 0} N_{k_n/k}U_n$, 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.

2020 Math. Subj. Class. 11R23, 11R37, 11R18, 11R34, 11R27, 11S25

Keywords: Tate module, Universal norm elements, cyclotomic $\mathbb{Z}_p$-extension, the Kuz'min–Gross conjecture.