Iwasawa theory for Artin representations I

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 $\rho$ of $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ 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 $\rho$ to the local Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$ has a 1-dimensional constituent $\varepsilon$ with multiplicity 1. If we fix the choice of such an $\varepsilon$, 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 $\mathbf{Z}_p$-extension of $\mathbf{Q}$ is a cotorsion module over the Iwasawa algebra $\Lambda$. That result is valid for an Artin representation of arbitrary dimension $d$ under the assumption that $d^{+} = 1$ and that such an $\varepsilon$ 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 $\rho$ by Hida theory.