\(T\)-ranks of Iwasawa modules
Let \(K\) be a number field and let \(L/K\) be a \({\mathbb Z}_p\)-extension. Let \(\Lambda:={\mathbb Z}_pT\cong {\mathbb Z}_p[[\mathrm{Gal}(L/K)]]\) be the Iwasawa algebra over \(L/K\). A projective limit \(A:=\lim_{\leftarrow n} A_n\) of finite modules is called a {\em Fukuda module} if there exist \(e\in {\mathbb N}\) and a submodule \(Y\subseteq A\) such that for \(n\geq e\) the canonical map \(A\to A_n\) is surjective and \(A_n\cong A/(\nu_{n,e}(T)Y)\), where \(\nu_{n,e}(T)=\frac{(T+1)^{p^n}-1}{(T+1)^{p^e}-1}\in{\mathbb Z}_p [T]\). The name Fukuda modules comes from a result of \textit{T. Fukuda} [Proc. Japan Acad., Ser. A 70, No. 8, 264--266 (1994; Zbl 0823.11064)]. The author considers four types of Fukuda modules. The {\em \(T\)-rank} of an Iwasawa module \(A\) is defined as \(\mathrm{rank}_T(A):=v_p\big(|A/(TA)|\big)\), whenever this is finite, which is equivalent to have that \(T\) does not divide the characteristic polynomial \(F_A(T)\) of \(A\). The first main result is that if \(A\) is a Fukuda module attached to a \({\mathbb Z}_p\)-extension \(L/K\) with \(e(A)=0\) and \(\mathrm{rank}_T(A_{n+1}) -\mathrm{rank}_T(A_{n}) =j\), then \(\mathrm{rank}_T(A_{n+r})-\mathrm{rank}_T (A_{n+r-1})\leq j\) for any \(r\geq 1\). This result is a special case of Theorem 2.2. The second main result is that if a number field \(K\) contains exactly two primes dividing \(p\), then Gross conjecture holds for \(K\). This result is part of Theorem 3.4. If \(p\) splits completely in \(K/{\mathbb Q}\), then Leopoldt's conjecture can be reduced to a similar problem of Theorem 3.4. In Section 3 some non-trivial cases of Leopoldt's conjecture are numerically verified. In Section 4, the semi-simplicity of Fukuda modules is studied.
Iwasawa algebra, \({\mathbb Z}_p\)-extensions
scientific article; zbMATH DE number 6987925
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