Greenberg’s conjecture for Dirichlet characters of order divisible by \(p\). (English) Zbl 1061.11062
For any even Dirichlet character \(\chi\) and any prime \(p\), it is conjectured that the Iwasawa \(\lambda\)-invariant \(\lambda_{p,\chi}\) of the \(\chi\)-part of the ideal class group is zero [R. Greenberg, Am. J. Math. 98, 263–284 (1976; Zbl 0334.12013)]. This is often called Greenberg’s conjecture.
Let \(p\) be an odd prime number. In [T. Tsuji, Trans. Am. Math. Soc. 355, No. 9, 3699–3714 (2003; Zbl 1038.11072)], the author gave sufficient conditions for \(\lambda_{p,\chi}\) to be zero.
In the paper under review, the author considers these conditions and shows under some assumptions that there exist infinitely many characters \(\chi\) of order divisible by \(p\) for which Greenberg’s conjecture is true (Propositions 5 and 6) by using Kida’s formula due to W. M. Sinnott [Compos. Math. 53, 3–17 (1984; Zbl 0545.12011)].
Let \(p\) be an odd prime number. In [T. Tsuji, Trans. Am. Math. Soc. 355, No. 9, 3699–3714 (2003; Zbl 1038.11072)], the author gave sufficient conditions for \(\lambda_{p,\chi}\) to be zero.
In the paper under review, the author considers these conditions and shows under some assumptions that there exist infinitely many characters \(\chi\) of order divisible by \(p\) for which Greenberg’s conjecture is true (Propositions 5 and 6) by using Kida’s formula due to W. M. Sinnott [Compos. Math. 53, 3–17 (1984; Zbl 0545.12011)].
Reviewer: Hisao Taya (Sendai)
References:
| [1] | Ferrero, B., and Washington, L. C.: The Iwasawa invariant \(\mu_p\) vanishes for abelian number fields. Ann. of Math., 109 , 377-395 (1979). · Zbl 0443.12001 · doi:10.2307/1971116 |
| [2] | Greenberg, R.: On the Iwasawa invariants of totally real number fields. Amer. J. Math., 98 , 263-284 (1976). · Zbl 0334.12013 · doi:10.2307/2373625 |
| [3] | Mazur, B., and Wiles, A.: Class fields of abelian extensions of \(\mathbf{Q}\). Invent. Math., 76 , 179-330 (1984). · Zbl 0545.12005 · doi:10.1007/BF01388599 |
| [4] | Ichimura, H., and Sumida, H.: On the Iwasawa invariants of certain real abelian fields II. Internat. J. Math., 7 , 721-744 (1996). · Zbl 0881.11075 · doi:10.1142/S0129167X96000384 |
| [5] | Iwasawa, K.: On \(\mathbf{Z}_\ell\)-extension of algebraic number fields. Ann. of Math., 98 , 246-326 (1973). · Zbl 0285.12008 · doi:10.2307/1970784 |
| [6] | Sinnott, W.: On \(p\)-adic \(L\)-functions and the Riemann-Hurwitz genus formula. Compositio Math., 53 , 3-17 (1984). · Zbl 0545.12011 |
| [7] | Tsuji, T.: On the Iwasawa \(\lambda\)-invariants of real abelian fields. (2000) (preprint). · Zbl 1038.11072 · doi:10.1090/S0002-9947-03-03357-9 |
| [8] | Tsuji, T.: On Iwasawa \(\lambda\)-invariants for odd Dirichlet characters (2001) (in preparation). · Zbl 0985.11514 |
| [9] | Washington, L. C.: Introduction to Cyclotomic Fields. Grad. Texts in Math., vol. 83, Springer, Berlin-Heidelberg-New York (1982). · Zbl 0484.12001 |
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