Iwasawa descent and co-descent for units modulo circular units. With an appendix by J.-R. Belliard.
Let \(F\) be a totally real abelian number field, \(p\) an odd prime number, \(F_\infty\) the cyclotomic \(\mathbb{Z}_p\)-extension of \(F\) and \(U_n\), \(C_n\) the group of units, respectively circular units, of its \(n\)th layer \(F_n\) (the fixed field of \(\Gamma_n\)). Define \(B_\infty= \varinjlim \mathbb{Z}_p\otimes_{\mathbb{Z}}U_n/C_n\) and \(Y_\infty= \varprojlim \mathbb{Z}_p\otimes_{\mathbb{Z}} U_n/C_n\), with connecting maps induced by extension and the norm map, respectively. This paper studies descent and co-descent for \(B_\infty\) and \(Y_\infty\), i.e., the kernels and cokernels of \(\mathbb{Z}_p\otimes U_n/C_n@>{j_n}>> (B_\infty)^{\Gamma_n}\) and \((Y_\infty)_{\Gamma_n}@>{h_n}>> \mathbb{Z}_p\otimes U_n/C_n\) in terms of module theoretic invariants attached to \(F_\infty/F\). If \(U_n/C_n\) is replaced by the ideal class group of \(F_n\) one is back in the situation of the classical main conjecture of Iwasawa theory; however, now things are more difficult because there is no Galois theoretic interpretation of \(U_n/C_n\) in terms of class field theory. We summarize two theorems (in part). If \(\varprojlim \mathbb{Z}_p\otimes C_n\) is \(\Lambda\)-free, then \(j_n\) is injective and \(\text{coker\,}j_n\), \(\ker h_n\) and \(\text{coker\,}h_n\) stabilize with respect to extension maps. Moreover, if \(p\nmid[F:\mathbb{Q}]\), a characterwise discussion of \(\mathbb{Z}_p\otimes U_n/C_n\) is possible. The tools applied in the paper are, besides the Mazur-Wiles Theorem (proof of the main conjecture of Iwasawa theory), an exact sequence of Sinnott's involving the Artin map and the Gross kernel of everywhere local universal norms and, assuming the \(\Lambda\)-freeness of \(\varprojlim \mathbb{Z}_p\otimes C_n\), a thorough study of the cohomology of units and circular units. Related work is [\textit{J.-R. Belliard}, Submodules of units in Iwasawa theory, Number theory, Besançon (2002; Zbl 1161.11394); \textit{J.-R. Belliard} and \textit{N.-Q.-D. Thong}, Nagoya Math. J. 177, 77--115 (2005; Zbl 1092.11043); \textit{R. Kučera} and \textit{J. Nekovár}, J. Algebra 171, No. 2, 457--472 (1995; Zbl 0824.11071); \textit{L. V. Kuz'min}, Math. USSR, Izv. 6(1972), 263--321 (1973); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 36, 267--327 (1972; Zbl 0231.12013); \textit{T. Nguyen Quang Do}, J. Number Theory (to appear)]. An appendix by J.-R. Belliard concerns the \(\Lambda\)-freeness condition on \(\varprojlim \mathbb{Z}_p\otimes C_n\).
scientific article; zbMATH DE number 5074490