VOL. 86 | 2020 Iwasawa theory for Artin representations I
Chapter Author(s) Ralph Greenberg, Vinayak Vatsal
Editor(s) Masato Kurihara, Kenichi Bannai, Tadashi Ochiai, Takeshi Tsuji
Adv. Stud. Pure Math., 2020: 255-301 (2020) DOI: 10.2969/aspm/08610255

Abstract

We introduce a natural way to define Selmer groups and p-adic L-functions for modular forms of weight 1. The corresponding Galois representation ρ of Gal(Q/Q) is a 2-dimensional Artin representation with odd determinant. Thus, the dimension d+ of the (+1)-eigenspace for complex conjugation is 1. Choose a prime p such that the restriction of ρ to the local Galois group Gal(Qp/Qp) has a 1-dimensional constituent ε with multiplicity 1. If we fix the choice of such an ε, we can define a Selmer group and a p-adic L-function. On the algebraic side, we prove that the Selmer group over the cyclotomic Zp-extension of Q is a cotorsion module over the Iwasawa algebra Λ. That result is valid for an Artin representation of arbitrary dimension d under the assumption that d+=1 and that such an ε can be chosen. On the analytic side, the corresponding complex L-function has no critical values and the definition of the p-adic L-function depends on deforming the Galois representation ρ by Hida theory.

Information

Published: 1 January 2020
First available in Project Euclid: 12 January 2021

Digital Object Identifier: 10.2969/aspm/08610255

Subjects:
Primary: 11F11 , 11R23 , 11R34

Keywords: Artin representations , Iwasawa theory

Rights: Copyright © 2020 Mathematical Society of Japan

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