Minimal Subgroups of ${\rm GL}_2(\mathbb{Z}_{S})$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910593114439680 |
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| author | Daniels, Harris Rouse, Jeremy |
| author_facet | Daniels, Harris Rouse, Jeremy |
| contents | Let $E$ be an elliptic curve over a number field $L$ and for a finite set $S$ of primes, let $ρ_{E,S} : {\rm Gal}(\overline{L}/L) \to {\rm GL}_{2}(\mathbb{Z}_{S})$ be the $S$-adic Galois representation. If $L \cap \mathbb{Q}(ζ_{n}) = \mathbb{Q}$ for all positive integers $n$ whose prime factors are in $S$, then $\det ρ_{E,S} : {\rm Gal}(\overline{L}/L) \to \mathbb{Z}_{S}^{\times}$ is surjective. We say that a finite index subgroup $H \subseteq {\rm GL}_{2}(\mathbb{Z}_{S})$ is minimal if $\det : H \to \mathbb{Z}_{S}^{\times}$ is surjective, but $\det : K \to \mathbb{Z}_{S}^{\times}$ is not surjective for any proper closed subgroup $K$ of $H$. We show that there are no minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{S})$ unless $S = \{ 2 \}$, while minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{2})$ are plentiful. We give models for all the genus $0$ modular curves associated to minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{2})$, and construct an infinite family of elliptic curves over imaginary quadratic fields with bad reduction only at $2$ and with minimal $2$-adic image. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11049 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal Subgroups of ${\rm GL}_2(\mathbb{Z}_{S})$ Daniels, Harris Rouse, Jeremy Number Theory Primary 11G05, Secondary 11F80, 14H52, 22E50 Let $E$ be an elliptic curve over a number field $L$ and for a finite set $S$ of primes, let $ρ_{E,S} : {\rm Gal}(\overline{L}/L) \to {\rm GL}_{2}(\mathbb{Z}_{S})$ be the $S$-adic Galois representation. If $L \cap \mathbb{Q}(ζ_{n}) = \mathbb{Q}$ for all positive integers $n$ whose prime factors are in $S$, then $\det ρ_{E,S} : {\rm Gal}(\overline{L}/L) \to \mathbb{Z}_{S}^{\times}$ is surjective. We say that a finite index subgroup $H \subseteq {\rm GL}_{2}(\mathbb{Z}_{S})$ is minimal if $\det : H \to \mathbb{Z}_{S}^{\times}$ is surjective, but $\det : K \to \mathbb{Z}_{S}^{\times}$ is not surjective for any proper closed subgroup $K$ of $H$. We show that there are no minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{S})$ unless $S = \{ 2 \}$, while minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{2})$ are plentiful. We give models for all the genus $0$ modular curves associated to minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{2})$, and construct an infinite family of elliptic curves over imaginary quadratic fields with bad reduction only at $2$ and with minimal $2$-adic image. |
| title | Minimal Subgroups of ${\rm GL}_2(\mathbb{Z}_{S})$ |
| topic | Number Theory Primary 11G05, Secondary 11F80, 14H52, 22E50 |
| url | https://arxiv.org/abs/2402.11049 |