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

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.