Advanced Studies in Pure Mathematics

On $p$-Adic Zeta Functions and Class Groups of $\mathbb{Z}_{p}$-Extensions of certain Totally Real Fields

Hisao Taya

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Abstract

Let $k$ be a totally real field and $p$ an odd prime number. We assume that $p$ splits completely in $k$ and also that Leopoldt's conjecture is valid for $k$ and $p$. In this note, focusing on Greenberg's conjecture, we will report on our recent results concerning $p$-adic special functions and ideal class groups in the cyclotomic ${\mathbb{Z}}_p$-extension of $k$.

Article information

Source
Class Field Theory – Its Centenary and Prospect, K. Miyake, ed. (Tokyo: Mathematical Society of Japan, 2001), 401-414

Dates
Received: 31 August 1998
First available in Project Euclid: 13 September 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1536853287

Digital Object Identifier
doi:10.2969/aspm/03010401

Mathematical Reviews number (MathSciNet)
MR1846468

Zentralblatt MATH identifier
1041.11071

Subjects
Primary: 11R23: Iwasawa theory
Secondary: 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11R29: Class numbers, class groups, discriminants

Keywords
${\mathbb{Z}}_p$-extensions Iwasawa invariants $p$-adic zetafunctions ideal class groups

Citation

Taya, Hisao. On $p$-Adic Zeta Functions and Class Groups of $\mathbb{Z}_{p}$-Extensions of certain Totally Real Fields. Class Field Theory – Its Centenary and Prospect, 401--414, Mathematical Society of Japan, Tokyo, Japan, 2001. doi:10.2969/aspm/03010401. https://projecteuclid.org/euclid.aspm/1536853287


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