On the image of the Galois representation associated to a non-CM Hida family
| Autor: | Jaclyn Lang |
|---|---|
| Rok vydání: | 2014 |
| Předmět: |
Pure mathematics
Hecke algebra 11F80 Absolutely irreducible Mathematics::Number Theory Modular form 01 natural sciences 11F85 Prime (order theory) 0103 physical sciences FOS: Mathematics Number Theory (math.NT) 0101 mathematics Galois representation Mathematics::Representation Theory Mathematics Algebra and Number Theory Mathematics - Number Theory Image (category theory) 010102 general mathematics Hida family 11F80 11F85 11F11 11F11 16. Peace & justice Subring Galois module Galois deformation 010307 mathematical physics Irreducible component |
| Zdroj: | Jaclyn Lang Algebra Number Theory 10, no. 1 (2016), 155-194 |
| DOI: | 10.48550/arxiv.1409.8651 |
| Popis: | Fix a prime $p > 2$. Let $\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{I})$ be the Galois representation coming from a non-CM irreducible component $\mathbb{I}$ of Hida's $p$-ordinary Hecke algebra. Assume the residual representation $\bar{\rho}$ is absolutely irreducible. Under a minor technical condition we identify a subring $\mathbb{I}_0$ of $\mathbb{I}$ containing $\mathbb{Z}_p[[T]]$ such that the image of $\rho$ is large with respect to $\mathbb{I}_0$. That is, $\text{Im} \rho$ contains $\text{ker}(\text{SL}_2(\mathbb{I}_0) \to \text{SL}_2(\mathbb{I}_0/\mathfrak{a}))$ for some non-zero $\mathbb{I}_0$-ideal $\mathfrak{a}$. This paper builds on recent work of Hida who showed that the image of such a Galois representation is large with respect to $\mathbb{Z}_p[[T]]$. Our result is an $\mathbb{I}$-adic analogue of the description of the image of the Galois representation attached to a non-CM classical modular form obtained by Ribet and Momose in the 1980s. Comment: 30 pages |
| Databáze: | OpenAIRE |
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