\(L_p(1,\chi)\bmod p\). (English) Zbl 1046.11080

Let \(k\) be a number field, \(\chi\) a Dirichlet character of conductor \(\Delta\), and \(p\) an odd prime coprime to \(\Delta\). The main result of this article is the congruence \[ \chi(p)^{-1} L_p(1,\chi) \equiv \sum_{t=1}^{p-1} (1 + \tfrac12 + \ldots + \tfrac1t) (\chi(rt) + \chi(rt+1) + \ldots + \chi(rt+r-1)) \bmod p, \] where \(r\) and \(s\) are integers with \(rp + s\Delta = 1\). If \(K = \mathbb Q(\sqrt{D}\,)\) is a real quadratic number field with associated Dirichlet character \(\chi\) in which \(p\) splits, and if the right hand side of the congruence above is nonzero, then the Iwasawa \(\mu\)- and \(\lambda\)-invariant of \(K\) vanish.


11R23 Iwasawa theory
11S40 Zeta functions and \(L\)-functions
Full Text: DOI Euclid


[1] Colmez, P.: Residu en \(s=1\) des functions zeta \(p\)-adiques. Invent. Math., 91 , 371-389 (1988). · Zbl 0651.12010 · doi:10.1007/BF01389373
[2] Greenberg, R.: On the Iwasawa invariants of totally real fields. Amer. J. Math., 98 , 263-284 (1976). · Zbl 0334.12013 · doi:10.2307/2373625
[3] Oh, J.: \(\ell\)-adic \(L\)-functions and rational function measures. Acta Arith., 83 , 369-379 (1998). · Zbl 0920.11073
[4] Taya, H.: On \(p\)-adic zeta functions and \(\mathbf{Z}_{p}\)-extensions of certain totally real number fields. Tohoku Math. J., 51 , 21-33 (1999). · Zbl 0943.11049 · doi:10.2748/tmj/1178224850
[5] Washington, L.: Introduction to Cyclotomic Fields. Grad. Texts in Math., 83, 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.