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

Takashi Hara, Tadashi Ochiai

TL;DR

This work constructs a multi-variable p-adic Artin L-function for Artin motives over CM fields by combining Katz–Hida–Tilouine p-adic Hecke L-functions with Brauer induction to handle arbitrary Artin representations. It develops a gluing mechanism across intermediate CM fields, matches local epsilon and period factors, and extends Davenport–Hasse type Gauss-sum identities to truncated Witt vector settings to control interpolation. The main results prove integrality of the p-adic Artin L-function in the Iwasawa algebra and establish (under a broad IMC-compatibility hypothesis) the Iwasawa main conjecture for these Artin representations in the CM setting. The approach generalizes Greenberg’s one-variable theory from totally real fields to CM fields and provides a framework for non-commutative Iwasawa theory via a careful descent of characteristic ideals and period–epsilon factor matching.

Abstract

For an algebraic Hecke character defined on a CM field $F$ of degree $2d$, Katz constructed a $p$-adic $L$-function of $d+1+δ_{F,p}$ variables in his innovative paper published in 1978, where $δ_{F,p}$ 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 $d+1+δ_{F,p}$ 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.

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

TL;DR

This work constructs a multi-variable p-adic Artin L-function for Artin motives over CM fields by combining Katz–Hida–Tilouine p-adic Hecke L-functions with Brauer induction to handle arbitrary Artin representations. It develops a gluing mechanism across intermediate CM fields, matches local epsilon and period factors, and extends Davenport–Hasse type Gauss-sum identities to truncated Witt vector settings to control interpolation. The main results prove integrality of the p-adic Artin L-function in the Iwasawa algebra and establish (under a broad IMC-compatibility hypothesis) the Iwasawa main conjecture for these Artin representations in the CM setting. The approach generalizes Greenberg’s one-variable theory from totally real fields to CM fields and provides a framework for non-commutative Iwasawa theory via a careful descent of characteristic ideals and period–epsilon factor matching.

Abstract

For an algebraic Hecke character defined on a CM field of degree , Katz constructed a -adic -function of variables in his innovative paper published in 1978, where denotes the Leopoldt defect for and . In the present article, we generalise the result of Katz under several technical conditions (containing the absolute unramifiedness of at ), and construct a -adic Artin -function of variables, which interpolates critical values of the Artin -function associated to a -unramified Artin representation of the absolute Galois group . 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.
Paper Structure (14 sections, 15 theorems, 137 equations)

This paper contains 14 sections, 15 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. The $p$-adic Hecke $L$-functions for CM fields
  3. General settings
  4. Period relations of CM periods
  5. Construction of the $p$-adic Hecke $L$-functions for CM fields
  6. Constructing the $p$-adic Artin $L$-function in the field of fractions of the Iwasawa algebra
  7. Local $L$-factors and local $\epsilon$-factors
  8. Artin representations cutting out CM fields
  9. Gluing $p$-adic Hecke $L$-functions
  10. Integrality of the $p$-adic Artin $L$-function and the Iwasawa main conjecture
  11. Iwasawa main conjectures and main results
  12. Algebraic preliminaries
  13. Integrality of the $p$-adic Artin $L$-function
  14. Davenport--Hasse relation over rings of truncated Witt vectors

Key Result

Theorem 1

Assume that $F$ is absolutely unramified at $(p)$ and take a branch character $\psi^\mathrm{gal}$ as above. Then there exists a unique element $L_{p, \Sigma_F}(\psi )$ of $\widehat{\mathcal{O}}^\mathrm{ur} [[\Gamma_{F,\, \max}]]$ satisfying with for each algebraic Hecke character $\eta\colon \mathbb{A}_F^\times/F^\times \rightarrow \overline{\mathbb{Q}}^\times$ satisfying the following two condi

Theorems & Definitions (42)

  • Theorem 1: =Theorem \ref{['thm:KHT']}, Katz, Hida--Tilouine
  • Theorem A: a part of Theorem \ref{['theorem:integrality_of_CMp-adicL_Artin']}
  • Theorem B: a part of Theorem \ref{['theorem:integrality_of_CMp-adicL_Artin']}
  • Definition 2.1
  • Remark 2.2: On CM periods
  • Lemma 2.3: Period relation, see also deShalit
  • Remark 2.4: multi-index notation
  • Theorem 2.5
  • proof
  • Remark 2.6: Independency of the polarisation parameter $\delta$
  • ...and 32 more