Examples of \(\lambda\)-invariants. (English) Zbl 0711.11039

The author studies the Iwasawa \(\lambda\)-invariant of the cyclotomic field of \(n\)-th roots of 1. As an application she calculates the values of the minus-part of \(\lambda =\lambda_ p\) for all odd primes \(p<100\) and all prime conductors \(n<100,\) \(n\neq p\); these are given in a table. The method rests on the author’s previous work [ibid. 64, No. 3, 359–375 (1989; Zbl 0689.12004)]. Her starting point is the decomposition of \(\lambda_ p^-\) as a sum of \(\lambda_ p(g_{\chi})\), the \(\lambda\)–invariants of the \(p\)-adic L-functions attached to the odd characters \(\chi\) of the field. A key position is occupied by a power series \({\hat \beta}{}_{\chi}(T)\) whose coefficients are not too hard to calculate and whose \(\lambda\)-invariant \(\lambda_ p({\hat \beta})\) has a close connection to \(\lambda_ p(g_{\chi})\). This connection was proved by the author [loc. cit.] under the assumption (always satisfied in the present cases) that \(\lambda_ p({\hat \beta})<p^ 2;\) a general proof has been provided by Y. Kida [Sci. Rep. Kanazawa Univ. 30, 33–38 (1985; Zbl 0595.12010)].
Reviewer: T.Metsänkylä


11R23 Iwasawa theory
11R18 Cyclotomic extensions
11R29 Class numbers, class groups, discriminants
11S40 Zeta functions and \(L\)-functions
Full Text: DOI EuDML


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