Iwasawa Theory for Artin Representations, I
| Autor: | Ralph Greenberg, Vinayak Vatsal |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Mathematics - Number Theory
Selmer group Mathematics::Number Theory Modular form Dimension (graph theory) Galois group 11F11 Iwasawa theory 16. Peace & justice Galois module Combinatorics 11R23 11R34 Iwasawa algebra FOS: Mathematics Number Theory (math.NT) Algebraic number Artin representations Mathematics |
| Zdroj: | Development of Iwasawa Theory — the Centennial of K. Iwasawa's Birth, M. Kurihara, K. Bannai, T. Ochiai and T. Tsuji, eds. (Tokyo: Mathematical Society of Japan, 2020) |
| DOI: | 10.48550/arxiv.1806.05659 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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