Let $E$ be an elliptic curve over a number field $L$ and for a finite set $S$ of primes, let $\rho _{E,S} : \mathrm{Gal}(\bar{L}/L) \rightarrow \mathrm{GL}_{2}(\mathbb{Z}_{S})$ be the $S$-adic Galois representation. If $L \cap \mathbb{Q}(\zeta _{n}) = \mathbb{Q}$ for all positive integers $n$ whose prime factors are in $S$, then $\det \rho _{E,S} : \mathrm{Gal}(\bar{L}/L) \rightarrow \mathbb{Z}_{S}^{\times }$ is surjective. We say that a finite index subgroup $H \subseteq \mathrm{GL}_{2}(\mathbb{Z}_{S})$ is minimal if $\det : H \rightarrow \mathbb{Z}_{S}^{\times }$ is surjective, but $\det : K \rightarrow \mathbb{Z}_{S}^{\times }$ is not surjective for any proper closed subgroup $K$ of $H$. We show that there are no minimal subgroups of $\mathrm{GL}_{2}(\mathbb{Z}_{S})$ unless $S = \lbrace 2 \rbrace $, while minimal subgroups of $\mathrm{GL}_{2}(\mathbb{Z}_{2})$ are plentiful. We give models for all the genus $0$ modular curves associated to minimal subgroups of $\mathrm{GL}_{2}(\mathbb{Z}_{2})$, and construct an infinite family of elliptic curves over imaginary quadratic fields with bad reduction only at $2$ and with minimal $2$-adic image.
Soient $L$ un corps de nombres, $E$ une courbe elliptique sur $L$, $S$ un ensemble de nombres premiers, et $\rho _{E,S} : \mathrm{Gal}(\bar{L}/L) \rightarrow \mathrm{GL}_{2}(\mathbb{Z}_{S})$ la représentation galoisienne $S$-adique. Si $L \cap \mathbb{Q}(\zeta _{n}) = \mathbb{Q}$ pour chaque entier $n$ dont tous les facteurs premiers sont dans $S$, alors $\det \rho _{E,S} : \mathrm{Gal}(\bar{L}/L) \rightarrow \mathrm{GL}_{2}(\mathbb{Z}_{S})$ est surjectif. Disons qu’un sous-groupe $H$ de $\mathrm{GL}_{2}(\mathbb{Z}_{S})$ d’indice fini est minimal si $\det : H \rightarrow \mathbb{Z}_{S}^{\times }$ est surjectif, mais $\det : K \rightarrow \mathbb{Z}_{S}^{\times }$ n’est pas surjectif pour chaque sous-groupe $K$ de $H$ propre et fermé. Nous montrons que $\mathrm{GL}_{2}(\mathbb{Z}_{S})$ n’admet un sous-groupe minimal que si $S = \lbrace 2 \rbrace $, et que dans $\mathrm{GL}_{2}(\mathbb{Z}_{2})$ il y en a plein. Nous donnons un modèle pour toute courbe modulaire de genre $0$ associée à un sous-groupe minimal, et construisons une famille infinie de courbes elliptiques sur des corps quadratiques imaginaires ayant mauvaise réduction seulement en $2$ et une image $2$-adique minimale.
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Keywords: Elliptic Curves, Galois Representations, Profinite Groups
Harris B. Daniels 1 ; Jeremy Rouse 2
CC-BY-ND 4.0
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author = {Harris B. Daniels and Jeremy Rouse},
title = {Minimal {Subgroups} of $\mathrm{GL}_2(\mathbb{Z}_{S})$},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {579--597},
year = {2025},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {37},
number = {2},
doi = {10.5802/jtnb.1333},
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Harris B. Daniels; Jeremy Rouse. Minimal Subgroups of $\mathrm{GL}_2(\mathbb{Z}_{S})$. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 579-597. doi: 10.5802/jtnb.1333
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