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

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.

43 pages, Title is changed a bit. Theorem 2.5 (and Theorem in Introduction) is renewed; we slightly modify Katz, Hida and Tilouine's measure and construct the p-adic Hecke L-function independent of auxiliary element delta. Correspondingly, the proof of Theorem 3.2 (construction and the interpolation formula of the p-adic Artin L-function) is revised