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 $k$.