×

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
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] [C1] Childress, N.,{\(\lambda\)}-Invariants and {\(\Gamma\)}-transforms, Manuscripta Math.64 (1989), 359–375 · Zbl 0689.12004
[2] [C2] Childress, N., ”Zeros ofp-acidL-functions,” Ph.D. Dissertation, The Ohio State University, 1985
[3] [D-F-K-S] Dummit, D., Ford, D., Kisilevsky, H., and Sands, J., ”Computation of Iwasawa Lambda Invariants for Imaginary Quadratic Fields,” (manuscript) · Zbl 0722.11052
[4] [E-M] Ernvall, R., and Metsänkylä, T.,A Method for computing the Iwasawa {\(\lambda\)}-invariant, Math. Comp.49 179(1987), 281–294 · Zbl 0601.12010
[5] [Go] Gold, R.,Examples of Iwasawa Invariants II, Acta Arith.26 (1975), 233–240 · Zbl 0264.12001
[6] [Gr] Greenberg, R.,On the Iwasawa invariants of totally real number fields, Amer. J. Math.98 (1976), 263–284 · Zbl 0334.12013
[7] [I] Iwasawa, K.,On p-adic L-functions, Ann. of Math.89 (1969), 198–205 · Zbl 0186.09201
[8] [L1] Lang, S., ”Cyclotomic Fields. Graduate Texts in Mathematics,” Springer-Verlag, New York, 1978
[9] [L2] Lang, S., ”Cyclotomic Fields II. Graduate Texts in Mathematics,” Spring-Verlag, New York, 1980.
[10] [S] Sinnott, W.,On the {\(\mu\)}-invariant of the {\(\Gamma\)}-transform of a rational function, Invent. Math.75 (1984), 273–282 · Zbl 0531.12004
[11] [W] Washington, L., ”Introduction to Cyclotomic Fields. Graduate Texts in Mathematics,” Springer-Verlag, New York, 1982. · Zbl 0484.12001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.