On the $p$-adic $L$-function and Iwasawa Main Conjecture for an Artin motive over a CM field

TL;DR

This paper generalizes Katz's construction of $p$-adic $L$-functions for algebraic Hecke characters over CM fields, extending it to Artin $L$-functions and addressing new technical challenges.

Contribution

It constructs a $p$-adic Artin $L$-function over CM fields that interpolates critical values, extending Greenberg's approach from totally real fields.

Findings

Constructed a $p$-adic Artin $L$-function with $d+1+ ext{Leopoldt defect}$ variables.

Generalized Katz's $p$-adic $L$-function to Artin representations over CM fields.

Addressed new difficulties arising in the CM field case.

Abstract

For an algebraic Hecke character defined on a CM field $F$ of degree $2d$, Katz constructed a $p$-adic $L$-function of $d+1+\delta_{F,p}$ variables in his innovative paper published in 1978, where $\delta_{F,p}$ denotes the Leopoldt defect for $F$ and $p$. In the present article, we generalise the result of Katz under several technical conditions (containing the absolute unramifiedness of $F$ at $p$), and construct a $p$-adic Artin $L$-function of $d+1+\delta_{F,p}$ variables, which interpolates critical values of the Artin $L$-function associated to a $p$-unramified Artin representation of the absolute Galois group $G_F$. Our construction is an analogue over a CM field of Greenberg's construction over a totally real field, but there appear new difficulties which do not matter in Greenberg's case.