Moscow Mathematical Journal
Volume 22, Issue 1, January–March 2022 pp. 121–132.
On Universal Norm Elements and a Problem of Coleman
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
Authors:
Soogil Seo (1)
Author institution:(1) Department of Mathematics, Yonsei University, 134 Sinchon-Dong, Seodaemun-Gu, Seoul 120-749, South Korea
Summary:
Keywords: Tate module, Universal norm elements, cyclotomic
$\mathbb{Z}_p$-extension, the Kuz'min–Gross conjecture.
Contents
Full-Text PDF