Effective bounds for adelic Galois representations attached to elliptic curves over the rationals
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866918359688282112 |
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| author | Furio, Lorenzo |
| author_facet | Furio, Lorenzo |
| contents | Given an elliptic curve $E$ defined over $\mathbb{Q}$ without complex multiplication, we provide an explicit sharp bound on the index of the image of the adelic representation $ρ_E$. In particular, if $\operatorname{h}_{\mathcal{F}}(E)$ is the stable Faltings height of $E$, we show that $[\operatorname{GL}_2(\widehat{\mathbb{Z}}) : \operatorname{Im}ρ_E]$ is bounded above by $10^{21} (\operatorname{h}_{\mathcal{F}}(E)+40)^{4.42}$, and, for $\operatorname{h}_{\mathcal{F}}(E)$ tending to infinity, by $\operatorname{h}_{\mathcal{F}}(E)^{3+o(1)}$. We also classify the possible (conjecturally non-existent) images of the representations $ρ_{E,p^n}$ whenever $\operatorname{Im}ρ_{E,p}$ is contained in the normaliser of a non-split Cartan. This result improves previous work of Zywina and Lombardo. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10340 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Effective bounds for adelic Galois representations attached to elliptic curves over the rationals Furio, Lorenzo Number Theory Algebraic Geometry 11G05, 11F80 Given an elliptic curve $E$ defined over $\mathbb{Q}$ without complex multiplication, we provide an explicit sharp bound on the index of the image of the adelic representation $ρ_E$. In particular, if $\operatorname{h}_{\mathcal{F}}(E)$ is the stable Faltings height of $E$, we show that $[\operatorname{GL}_2(\widehat{\mathbb{Z}}) : \operatorname{Im}ρ_E]$ is bounded above by $10^{21} (\operatorname{h}_{\mathcal{F}}(E)+40)^{4.42}$, and, for $\operatorname{h}_{\mathcal{F}}(E)$ tending to infinity, by $\operatorname{h}_{\mathcal{F}}(E)^{3+o(1)}$. We also classify the possible (conjecturally non-existent) images of the representations $ρ_{E,p^n}$ whenever $\operatorname{Im}ρ_{E,p}$ is contained in the normaliser of a non-split Cartan. This result improves previous work of Zywina and Lombardo. |
| title | Effective bounds for adelic Galois representations attached to elliptic curves over the rationals |
| topic | Number Theory Algebraic Geometry 11G05, 11F80 |
| url | https://arxiv.org/abs/2412.10340 |