## Advanced Studies in Pure Mathematics

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

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

#### 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