## 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

### MSC:

 11R23 Iwasawa theory 11R18 Cyclotomic extensions 11R29 Class numbers, class groups, discriminants 11S40 Zeta functions and $$L$$-functions

### Citations:

Zbl 0689.12004; Zbl 0595.12010
Full Text:

### References:

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