---
abstract: 'Let $\Gamma$ be a finite rank subgroup of the linear torus or an elliptic curve defined over a number field with complex multiplication. We prove that the group of points which are rational over the field generated by all elements in the divisible hull of $\Gamma$, is free abelian modulo this divisible hull. This proves that a necessary condition for Rémond’s generalized Lehmer conjecture is satisfied.'
address: 'Fakultät für Mathematik, Universität Duisburg-Essen, 45117 Germany'
author:
- Lukas Pottmeyer
title: Fields Generated by Finite Rank Subgroups of Tori and Elliptic Curves
---

Introduction
============

We fix once and for all an algebraic closure $\overline{\mathbb{Q}}$ of the rational numbers and assume that all algebraic extensions of $\mathbb{Q}$ are contained in this closure. The absolute logarithmic Weil-height $h$ on $\overline{\mathbb{Q}}$ can be defined as follows: For $\alpha_1\in\overline{\mathbb{Q}}$ let $f(x)=a_d (x-\alpha_1)\cdot\ldots\cdot(x-\alpha_d)\in\mathbb{Z}[x]$ be irreducible, then $$h(\alpha_1)=\frac{1}{d}\log\left( \vert a_d \vert \cdot \prod_{i=1}^d \max\{1,\vert \alpha_i\vert \}\right).$$ This function satisfies $h(\alpha^r)=\vert r \vert h(\alpha)$ for all $\alpha \in \overline{\mathbb{Q}}^*$ and all $r\in\mathbb{Q}$, and vanishes precisely at $0$ and roots of unity. It follows that for any $\alpha \in \overline{\mathbb{Q}}^*$ which is not a root of unity, the sequence $\alpha^{\nicefrac{1}{n}}$, $n\in\mathbb{N}=\{1,2,3,\ldots\}$, contains algebraic numbers of arbitrarily small positive height.

Let us denote the set of roots of unity by $\mu$. We can ask the following question: Are there elements of arbitrarily small positive height in $$\mathbb{Q}(\mu,\alpha, \alpha^{\nicefrac{1}{2}}, \alpha^{\nicefrac{1}{3}},\ldots)^*$$ which are not of the form $\zeta \cdot \alpha^q$ for some $\zeta \in \mu$ and some $q\in\mathbb{Q}$? Conjecturally the answer is “no”. To formulate this conjecture in full generality, we need some further notation.

Let ${\mathcal{G}}=A\times \mathbb{G}_m^N$ for some $N\in\mathbb{N}_0$ and an abelian variety $A$ defined over a number field $K$ equipped with an ample symmetric line bundle $\mathcal{L}$. The choice of this line bundle defines a Néron-Tate height $h_{\mathcal{L}}$ on $A(\overline{\mathbb{Q}})$.

The canonical height $\widehat{h}_{{\mathcal{G}}}$ on ${\mathcal{G}}(\overline{\mathbb{Q}})$ is given as the sum of the Néron-Tate height and the Weil-height on each component. This means, for $(P,\alpha_1,\ldots,\alpha_N)\in{\mathcal{G}}(\overline{\mathbb{Q}})$ we set $$\widehat{h}_{{\mathcal{G}}}(P,\alpha_1,\ldots,\alpha_N)=h_{\mathcal{L}}(P)+\sum_{i=1}^N h(\alpha_i).$$ For definitions, properties and applications of these height functions we refer to [@BG].

If $G$ is any divisible group with a subgroup $\Gamma$, then we define the *divisible hull* of $\Gamma$ to be the group $$\Gamma_{\operatorname{div}}:=\{\gamma \in G \vert n\gamma \in \Gamma \text{ for some } n \in \mathbb{N}\}.$$

Let $\Gamma$ be a subgroup of ${\mathcal{G}}(\overline{\mathbb{Q}})$. Then we denote by $\operatorname{End}({\mathcal{G}})\cdot \Gamma$ the subgroup of ${\mathcal{G}}(\overline{\mathbb{Q}})$ generated by all elements of the form $\varphi(\gamma)$ with $\varphi\in\operatorname{End}({\mathcal{G}})=\operatorname{End}_{\overline{\mathbb{Q}}}({\mathcal{G}})$ and $\gamma\in\Gamma$. Moreover we define $$\Gamma_{\operatorname{sat}}:= (\operatorname{End}({\mathcal{G}})\cdot \Gamma)_{\operatorname{div}}.$$ Note that $\operatorname{End}({\mathcal{G}})\cdot \Gamma = \Gamma$ if $\operatorname{End}({\mathcal{G}})=\mathbb{Z}$. In this case $\Gamma_{\operatorname{div}} = \Gamma_{\operatorname{sat}}$. Note that in all cases $\Gamma_{\operatorname{sat}}$ is a divisible group of finite rank, whenever the rank of $\Gamma$ is finite. By the *rank* of $\Gamma$, we mean the rational rank; this is the maximal number of linearly independent elements in $\Gamma$. Moreover, if $\Gamma$ has rank zero then $\Gamma_{\operatorname{sat}}=\Gamma_{\operatorname{div}}$ is precisely given by the torsion subgroup ${\mathcal{G}}_{\operatorname{tors}}$. Now we can formulate Rémond’s generalized Lehmer conjecture [@Re11], Conjecture 3.4.

\[conj\] Let ${\mathcal{G}}$ be either a torus or an abelian variety and let $\Gamma$ be a finite rank subgroup of ${\mathcal{G}}(\overline{\mathbb{Q}})$. An element $\alpha \in {\mathcal{G}}(\overline{\mathbb{Q}})$ is called $\Gamma$-transversal, if it is contained in some translate $\gamma + B$, where $\gamma\in\Gamma_{\operatorname{sat}}$ and $B$ is a connected proper algebraic subgroup of ${\mathcal{G}}$.

-   There exists a positive constant $c$ such that $$\widehat{h}_{{\mathcal{G}}}(\alpha)\geq \frac{c}{[K(\Gamma_{\operatorname{sat}})(\alpha):K(\Gamma_{\operatorname{sat}})]^{\nicefrac{1}{\dim({\mathcal{G}})}}} \quad \forall ~  \alpha \in {\mathcal{G}}(\overline{\mathbb{Q}}) \text{ which are not } \Gamma\text{-transversal}.$$

-   For any $\varepsilon>0$ there is a positive constant $c_{\varepsilon}$ such that $$\widehat{h}_{{\mathcal{G}}}(\alpha)\geq \frac{c_{\varepsilon}}{[K(\Gamma_{\operatorname{sat}})(\alpha):K(\Gamma_{\operatorname{sat}})]^{\nicefrac{1}{\dim({\mathcal{G}})}+\varepsilon}} \quad \forall ~  \alpha \in {\mathcal{G}}(\overline{\mathbb{Q}}) \text{ which are not } \Gamma\text{-transversal}.$$

-   For all finite extensions $L/K(\Gamma_{\operatorname{sat}})$ there is a positive constant $c_L$ such that $$\widehat{h}_{{\mathcal{G}}}(\alpha)\geq c_L \quad \forall ~ \alpha \in {\mathcal{G}}(L)\setminus \Gamma_{\operatorname{sat}}.$$

Conjecture \[conj\] is weaker than Rémond’s original conjecture in two points. Firstly, Conjecture 3.4 from [@Re11] predicts lower bounds for the height of subvarieties of ${\mathcal{G}}$, not just for the height of points. Secondly, the exponent on the right hand side of (a) and (b) is smaller than our exponents $\frac{1}{\dim({\mathcal{G}})}$, resp. $\frac{1}{\dim({\mathcal{G}})}+\varepsilon$. Since the focus of this paper lies solely on part (c), we did not introduce the necessary notation to present the exponents conjectured by Rémond. On the other hand, this conjecture could be generalized to cover also semi-abelian varieties of the form $A\times \mathbb{G}_m^N$. For part (c) this generalization of the conjecture can be found in [@Pl19], Conjecture 1.2.

Obviously, part (a) of Conjecture \[conj\] implies part (b). It is also true that part (b) implies part (c). If $\dim({\mathcal{G}})=1$, this follows since in this case $\alpha$ is $\Gamma$-transversal if and only if $\alpha \in \Gamma_{\operatorname{sat}}$. For general ${\mathcal{G}}$ the implication (b) $\Rightarrow$ (c) follows as a very special case from the strong result [@Re11], Theorem 3.7.

Part (a) of Conjecture \[conj\] is much stronger than the famous Lehmer conjecture which has its origin in [@Le33]. This conjecture predicts the existence of a positive constant $c$ such that $h(\alpha)\geq \frac{c}{[\mathbb{Q}(\alpha):\mathbb{Q}]}$ for all $\alpha \in \overline{\mathbb{Q}}^*\setminus \mu$.

There are some results on this conjecture in the case that the rank of $\Gamma$ is zero. Delsinne [@Del09] proved Conjecture \[conj\] (b) in the case that ${\mathcal{G}}=\mathbb{G}^N$ and $\Gamma_{\operatorname{sat}}={\mathcal{G}}_{\operatorname{tors}}$. Also under the assumption $\Gamma_{\operatorname{sat}}={\mathcal{G}}_{\operatorname{tors}}$, Carrizosa [@Ca09] proved Conjecture \[conj\] (b) in the case that ${\mathcal{G}}$ is an abelian variety with complex multiplication. (Previously, Delsinne’s result for $N=1$ has been proven in [@AZ00], and part (c) for ${\mathcal{G}}$ an abelian variety with complex multiplication and $\Gamma_{\operatorname{sat}}={\mathcal{G}}_{\operatorname{tors}}$ has been proven in [@BS04]).

Amoroso [@Am14] could achieve the following result towards the seemingly most easiest case of a group of positive rank: Let $\Gamma=\langle 2 \rangle$ be the subgroup of ${\mathcal{G}}=\mathbb{G}_m$ generated by $2$, and define $\Gamma_{3\operatorname{div}}=\{\alpha \in \mathbb{G}_m \vert 3^n \cdot \alpha \in \Gamma\}$. Then there is an effective constant $c>0$ such that $h(\alpha)\geq c$ for all $\alpha \in \mathbb{G}_m(\mathbb{Q}(\Gamma_{3\operatorname{div}}))\setminus \Gamma_{3\operatorname{div}}$. Under certain technical restrictions, a similar result for groups ${\mathcal{G}}=A\times \mathbb{G}_m$ and $\Gamma=\{0\} \times \langle b \rangle$, where $A$ is an elliptic curve and $b$ is an integer, has recently been announced in [@Pl19].

We will give some group theoretic support for the validity of Conjecture \[conj\] (c). In the next section we will see that Conjecture \[conj\] (c) implies that the group $\nicefrac{{\mathcal{G}}(L)}{\Gamma_{\operatorname{sat}}}$ is free abelian for all finite extensions $L/K(\Gamma_{\operatorname{sat}})$. Hence, the following theorem shows that a necessary condition for the truth of Conjecture \[conj\] (c) is fulfilled.

\[thm:freeab\] Let ${\mathcal{G}}$ be either $\mathbb{G}_m$ or an elliptic curve with complex multiplication defined over a number field $K$, and let $\Gamma$ be a subgroup of ${\mathcal{G}}(\overline{\mathbb{Q}})$ of finite rank. Then the group $\nicefrac{{\mathcal{G}}(L)}{\Gamma_{\operatorname{sat}}}$ is free abelian for all finite extensions $L/K(\Gamma_{\operatorname{sat}})$.

If the rank of $\Gamma$ is zero (i.e. if $\Gamma_{\operatorname{sat}}={\mathcal{G}}_{\operatorname{tors}}$), then the statement of Theorem \[thm:freeab\] is true for all semi-abelian varieties of the form $A\times \mathbb{G}_m^N$. This was proved by Bays, Hart and Pillay in the appendix of [@BHP]. Their result is used in our proof, as it provides a kind of *base case* (see Proposition \[prop:firstclaim\]).

The paper is organized as follows. In Section 2 we will clarify the connection between Rémond’s conjecture and free abelian groups. In Section 3 we recall a criterion due to Pontryagin for a group to be free abelian. We apply this criterion to show that the group $\nicefrac{{\mathcal{G}}(L)}{\Gamma_{\operatorname{sat}}}$ is free abelian if the torsion group of $$\label{eq:gr}
\nicefrac{{\mathcal{G}}(L)}{{\mathcal{G}}(K({\mathcal{G}}_{\operatorname{tors}}))+\Gamma_{\operatorname{sat}}} \quad \text{ or } \quad \nicefrac{{\mathcal{G}}(L)}{{\mathcal{G}}(K)+\Gamma_{\operatorname{sat}}}$$ has finite exponent (this reduction step is true in the general case ${\mathcal{G}}=A\times \mathbb{G}_m^N$). Next we prove Theorem \[thm:freeab\] in the case that ${\mathcal{G}}$ is the linear torus, by showing that the first group from is torsion free. This follows quite elementary by basic facts on cyclic field extensions.

We collect some facts on elliptic curves in Section 4. The proof that the exponent of the torsion part of the second group from is finite if ${\mathcal{G}}$ is an elliptic curve with complex multiplication is proven in Sections 6. The final Section 7 provides a result towards a proof of Theorem \[thm:freeab\] for ${\mathcal{G}}$ an elliptic curves without complex multiplication. In this case we prove that for all but finitely many prime numbers $\ell$ the second group from has trivial $\ell$-torsion.

In the case where ${\mathcal{G}}$ is an elliptic curve, we use some Kummer theory. In particular, we need that the rank of the Galois group of $K({\mathcal{G}}_{\operatorname{tors}},\frac{1}{\ell^m}\Gamma)/K({\mathcal{G}}_{\operatorname{tors}})$ is as large as possible for all primes $\ell$ and some integer $m$ which is equal to $1$ for all but finitely many primes. For ${\mathcal{G}}_{\operatorname{tors}}$ replaced by the $\ell$-torsion of ${\mathcal{G}}$ this result is due to Bashmakov ([@Ba], Theorem 6). As we could not find a reference for the precise statement used in this paper, we will present a proof of this result following the outline of V.§5 of [@La]. A main ingredient in this proof is Serre’s famous open image theorem [@Se].

Some group theory and Rémond’s Lemma
====================================

All abelian groups in this section will be written additively. A *norm* on an abelian group $G$ is a function ${\lVert\cdot\rVert}: G \longrightarrow \mathbb{R}_{\geq 0}$ satisfying

(i) ${\lVertg\rVert} = 0 ~ \Longleftrightarrow ~ g = 0$ is the neutral element,

(ii) ${\lVert g + f\rVert} \leq {\lVert g \rVert} + {\lVert f \rVert}$ for all $g,f \in G$, and

(iii) ${\lVert n\cdot g \rVert} = \vert n \vert \cdot {\lVert g \rVert}$ for all $g\in G$ and all $n \in \mathbb{Z}$.

By (i) and (iii), a group norm can only exist for torsion-free groups. If ${\lVert\cdot\rVert}$ only satisfies (ii) and (iii), then it is called a *semi-norm*. A norm ${\lVert\cdot\rVert}$ is called *discrete* on $G$ if and only if $0$ is not an accumulation point in the set $\{{\lVertg\rVert} \vert g\in G\}$.

One of the main properties of $\widehat{h}_{{\mathcal{G}}}$ is that it is well-defined on $\nicefrac{{\mathcal{G}}(\overline{\mathbb{Q}})}{{\mathcal{G}}_{\operatorname{tors}}}$, where ${\mathcal{G}}_{\operatorname{tors}}$ is the torsion subgroup of ${\mathcal{G}}$. Moreover, the map $$(P,\alpha_1,\ldots,\alpha_N) \mapsto \sqrt{h_{\mathcal{L}}(P)}+\sum_{i=1}^N h(\alpha_i)$$ is a norm on $\nicefrac{{\mathcal{G}}(\overline{\mathbb{Q}})}{{\mathcal{G}}_{\operatorname{tors}}} = \nicefrac{A(\overline{\mathbb{Q}})\times\mathbb{G}_m^N(\overline{\mathbb{Q}})}{{\mathcal{G}}_{\operatorname{tors}}}$. We will denote this norm on $\nicefrac{{\mathcal{G}}(\overline{\mathbb{Q}})}{{\mathcal{G}}_{\operatorname{tors}}}$ by $\Vert \cdot \Vert_h$.

Let ${\lVert.\rVert}$ be a norm on a divisible group $G$ and let $\Gamma$ be a subgroup of $G$. Then we define the function $${\lVert.\rVert}_{\Gamma}: \nicefrac{G}{\Gamma_{\operatorname{div}}} \longrightarrow \mathbb{R}_{\geq0} \quad ; \quad {\lVert[\alpha]\rVert}_{\Gamma}= \inf\{{\lVert\alpha+\gamma\rVert} \vert \gamma \in \Gamma_{\operatorname{div}}\} .$$ Since this map is well-defined we will simply write ${\lVert\alpha\rVert}_{\Gamma}$ for ${\lVert[\alpha]\rVert}_{\Gamma}$.

The following lemma is along the lines of [@Re11], Lemma 3.5. The proof is also due to Gaël Rémond, who presented a special case of it at the workshop on “Heights in Diophantine geometry, group theory and additive combinatorics” at the ESI in Vienna in November 2013. See also [@Gr17] for a quantitative version in the $G=\mathbb{G}_m$ case.

\[Bogomolov\] Let $G$ be a divisible group such that there is a norm ${\lVert.\rVert}$ on $\nicefrac{G}{G_{\operatorname{tors}}}$. Let $\Gamma$ be a subgroup of $G$ of rank $r <\infty$ with $\gamma_1,\dots,\gamma_r \in \Gamma$ linearly independent. Moreover, let $H$ be another subgroup of $G$ with $\gamma_1,\dots,\gamma_r \in H$ and such that ${\lVertg\rVert}\geq \kappa$ for all $g\in \nicefrac{H}{H_{\operatorname{tors}}}\setminus \nicefrac{\Gamma_{\operatorname{div}}\cap H}{H_{\operatorname{tors}}}$, for some constant $\kappa >0$. Then there exists a positive constant $c$ only depending on $H$ and $\Gamma$ such that ${\lVertg\rVert}_{\Gamma} \geq c$ for all $g\in H\setminus \Gamma_{\operatorname{div}}$.

Let $g \in H\setminus \Gamma_{\operatorname{div}}$, and let $\underline{a}=(a_1,\dots,a_r)$ and $\underline{b}=(b_1,\dots,b_r)$ be elements in $\mathbb{Q}^r$. Then properties (ii) and (iii) of ${\lVert.\rVert}$ yields $$\begin{aligned}
\label{lipschitz} \big\vert {\lVertg+a_1 \gamma_1+ \dots + a_r\gamma_r\rVert} - {\lVertg + b_1 \gamma_1 + \dots + b_r \gamma_r\rVert} \big\vert &\leq {\lVert(a_1 - b_1) \gamma_1 + \dots + (a_r - b_r)\gamma_r\rVert} \nonumber \\ &\leq \max_{1\leq i \leq r} \{{\lVert\gamma_i\rVert}\} \cdot \sum_{i=1}^r \vert a_i - b_i \vert.\end{aligned}$$ We have to bound ${\lVertg + b_1 \gamma_1 + \dots + b_r \gamma_r\rVert}$ from below, independently of $\underline{b}$. If $\underline{a}\in \frac{1}{m}\mathbb{Z}^{r}$, then $mg+ ma_1 \gamma_1 + \dots + ma_r \gamma_r\in H$ and ${\lVertg+ a_1 \gamma_1 + \dots + a_r \gamma_r\rVert} = \frac{1}{m}{\lVertmg + ma_1\gamma_1 + \dots + ma_r \gamma_r\rVert}$. Since $g$ is not an element in $\Gamma_{\operatorname{div}}$, we can apply our assumption on ${\lVert.\rVert}$ restricted to $\nicefrac{H}{H_{\operatorname{tors}}}\setminus\nicefrac{H \cap \Gamma_{\operatorname{div}}}{H_{\operatorname{tors}}}$ to conclude $$\label{bound}
{\lVertg + a_1 \gamma_1 + \dots + a_r \gamma_r\rVert} \geq \frac{\kappa}{m},$$ Let again $\underline{b}\in \mathbb{Q}^r$ be arbitrary and let $Q\geq 2r\max_{1\leq i \leq r} {\lVert\gamma_i\rVert} \kappa^{-1}$ be an integer. By Dirichlet’s approximation theorem, there is a positive integer $m$ and $\underline{a}\in\frac{1}{m}\mathbb{Z}^r$ such that $$\label{fin}
 m\leq Q^r \text{ and } \vert a_i - b_i \vert \leq \frac{1}{mQ} \text{ for all } i \in \{1,\ldots,r\}.$$ Combining and yields that ${\lVertg + b_1 \gamma_1 + \dots + b_r \gamma_r\rVert}$ is bounded from below by $$\begin{aligned}
 \geq &{\lVertg + a_1 \gamma_1+ \dots + a_r \gamma_r\rVert} - \big\vert {\lVertg+a_1 \gamma_1+ \dots + a_r\gamma_r\rVert} - {\lVertg + b_1 \gamma_1 + \dots + b_r \gamma_r\rVert} \big\vert \\
\geq & \frac{\kappa}{m} - \max_{1\leq i \leq r} \{{\lVert\gamma_i\rVert}\} \cdot \sum_{i=1}^n \vert a_i - b_i \vert\overset{\eqref{fin}}{\geq} \frac{\kappa}{m} - \frac{\max_{1\leq i \leq r} \{{\lVert\gamma_i\rVert}\} \cdot r}{mQ}\\  = & \frac{\kappa Q - \max_{1\leq i \leq r} \{{\lVert\gamma_i\rVert}\} \cdot r}{mQ} \geq \frac{\max_{1\leq i \leq r} \{{\lVert\gamma_i\rVert}\} \cdot r}{Q^{r+1}}.\end{aligned}$$ The latter is the postulated positive constant which only depends on $H$ and $\Gamma$.

As a corollary we will state the statement explicitly for $G={\mathcal{G}}(\overline{\mathbb{Q}})$, where ${\mathcal{G}}=A\times \mathbb{G}_m^N$ is a semi-abelian variety, and $H={\mathcal{G}}(F)$ for some field $F\subseteq\overline{\mathbb{Q}}$.

\[cor:mcG\] Let $\Gamma \subseteq {\mathcal{G}}(\overline{\mathbb{Q}})$ be a subgroup of finite rank, and let $\Gamma_{\operatorname{sat}}=\langle \gamma_1,\ldots,\gamma_r\rangle_{\operatorname{div}}$. Let $F$ be a subfield of $\overline{\mathbb{Q}}$ satisfying

(i) $\gamma_1,\ldots,\gamma_r \in {\mathcal{G}}(F)$, and

(ii) there is a positive constant $\kappa$ such that $\widehat{h}_{{\mathcal{G}}}(\alpha) \geq \kappa$ for all $\alpha \in {\mathcal{G}}(F)\setminus \Gamma_{\operatorname{sat}}$.

Then there is a positive constant $c$ only depending on $F$ and $\Gamma$ such that ${\lVert\alpha\rVert}_{h,\Gamma}\geq c$ for all $\alpha \in {\mathcal{G}}(F)\setminus \Gamma_{\operatorname{sat}}$.

\[heightisnorm\] Let $\Gamma$ be a subgroup of ${\mathcal{G}}(\overline{\mathbb{Q}})$ and ${\lVert.\rVert}$ a norm on $\nicefrac{{\mathcal{G}}(\overline{\mathbb{Q}})}{{\mathcal{G}}_{\operatorname{tors}}}$. Then the function ${\lVert.\rVert}_{\Gamma}$ is a seminorm on $\nicefrac{{\mathcal{G}}(\overline{\mathbb{Q}})}{\Gamma_{\operatorname{sat}}}$. If $\Gamma$ has finite rank, then the particular function ${\lVert.\rVert}_{h,\Gamma}$ is a norm.

First we will show that ${\lVert.\rVert}_{\Gamma}$ is a semi-norm, without additional assumptions on the group $\Gamma$. In order to do so, we have to check the properties (ii) and (iii) from the beginning of this section. These properties follow from the respective properties of the norm ${\lVert.\rVert}$.

The triangular inequality follows from $$\begin{aligned}
 {\lVert\alpha\rVert}_{\Gamma}+{\lVert\beta\rVert}_{\Gamma} &=\inf\{{\lVert\alpha+\gamma\rVert} \vert \gamma \in \Gamma_{\operatorname{div}}\}+\inf\{{\lVert\beta+\gamma'\rVert} \vert \gamma' \in \Gamma_{\operatorname{div}}\}\\
 &=\inf\{{\lVert\alpha+\gamma\rVert} + {\lVert\beta+\gamma'\rVert} \vert \gamma,\gamma' \in \Gamma_{\operatorname{div}}\}\\
 &\geq \inf\{{\lVert\alpha+\beta+\gamma+\gamma'\rVert} \vert \gamma,\gamma' \in \Gamma_{\operatorname{div}}\} \\
 &=\inf\{{\lVert\alpha+\beta+\gamma''\rVert} \vert \gamma'' \in \Gamma_{\operatorname{div}}\} = {\lVert\alpha+\beta\rVert}_{\Gamma},\end{aligned}$$ and the last statement follows similarly from the equation $$\begin{aligned}
 {\lVertn\alpha\rVert}_{\Gamma} &=\inf\{{\lVertn\alpha + \gamma\rVert} \vert \gamma \in \Gamma_{\operatorname{div}}\} =\inf\{{\lVertn(\alpha+\gamma')\rVert} \vert \gamma' \in \Gamma_{\operatorname{div}}\}\\
 &=\inf\{n{\lVert\alpha+\gamma'\rVert} \vert \gamma' \in \Gamma_{\operatorname{div}}\} =n{\lVert\alpha\rVert}_{\Gamma}.\end{aligned}$$ From now on we assume that $\Gamma$ is of rank $r<\infty$ and that the norm ${\lVert.\rVert}={\lVert.\rVert}_h$ is induced by the canonical height $\widehat{h}_{{\mathcal{G}}}$ on ${\mathcal{G}}$. Let $\gamma_1,\dots,\gamma_r$ be linearly independent elements in $\Gamma$. We are left to prove property (i); i.e. ${\lVert\alpha\rVert}_{h,\Gamma}=0 \Longleftrightarrow \alpha\in \Gamma_{\operatorname{sat}}$. Obviously it is ${\lVert\alpha\rVert}_{h,\Gamma}=0$ for all $\alpha \in \Gamma_{\operatorname{sat}}$. By Northcott’s theorem, ${\lVert.\rVert}_{h}$ is discrete on $\nicefrac{{\mathcal{G}}(F)}{{\mathcal{G}}_{\operatorname{tors}}(F)}$ for all number fields $F$. Hence, if $\alpha \in {\mathcal{G}}(\overline{\mathbb{Q}})\setminus \Gamma_{\operatorname{sat}}$ is arbitrary we set $F=\mathbb{Q}(\gamma_1,\dots,\gamma_r,\alpha)$ and apply Corollary \[cor:mcG\]. This yields ${\lVert\alpha\rVert}_{h,\Gamma} \neq 0$ and concludes the proof.

We use the notation from Conjecture \[conj\]. Since $\Gamma$ and $\operatorname{End}({\mathcal{G}})$ are of finite rank, the same is true for $\Gamma_{\operatorname{sat}}$. Hence, Lemma \[heightisnorm\] tells us that ${\lVert\cdot\rVert}_{h,\Gamma_{\operatorname{sat}}}$ is a norm on $\nicefrac{{\mathcal{G}}(L)}{\Gamma_{\operatorname{sat}}}$. This norm is discrete if and only if the statement of Conjecture \[conj\] (c) is true. Therefore, Conjecture \[conj\] (c) is true if and only if ${\lVert\cdot\rVert}_{h,\Gamma_{\operatorname{sat}}}$ is a discrete norm on $\nicefrac{{\mathcal{G}}(L)}{\Gamma_{\operatorname{sat}}}$.

Note that there exists a discrete norm on an abelian group if and only if this group is free abelian. This result was proved independently by Lawrence [@La84] and Zorzitto [@Zo85] for countable groups, and by Steprāns [@St85] in the general case. As this result is the bridge between Conjecture \[conj\] and our main theorem, we state it as a proposition.

\[LSZ\] An abelian group $G$ is free if and only if there is a discrete norm on $G$.

We conclude

$$\begin{aligned}
\text{Conjecture \ref{conj} (c) is true } &\Longleftrightarrow ~ {\lVert\cdot\rVert}_{h,\Gamma_{\operatorname{sat}}} \text{ is a discrete norm on } \nicefrac{{\mathcal{G}}(L)}{\Gamma_{\operatorname{sat}}}\\
 &\Longrightarrow ~ \nicefrac{{\mathcal{G}}(L)}{\Gamma_{\operatorname{sat}}} \text{ is free abelian}\end{aligned}$$

Hence, Theorem \[thm:freeab\] tells us, that at least there exists *some* discrete norm on $\nicefrac{{\mathcal{G}}(L)}{\Gamma_{\operatorname{sat}}}$ if ${\mathcal{G}}$ is either the linear torus or an elliptic curve with complex multiplication.

Pontryagin’s criterion
======================

The results of this section are valid in a more general setting, than needed for the proof of Theorem \[thm:freeab\]. Hence, in this section ${\mathcal{G}}= A\times \mathbb{G}_m^N$ is defined over a number field $K$, where $A$ is an abelian variety and $N \in \mathbb{N}_0$.

\[freemodgamma\] Let $F$ be any subfield of $K(\Gamma_{\operatorname{sat}})$. The group $\nicefrac{{\mathcal{G}}(K(\Gamma_{\operatorname{sat}}))}{\Gamma_{\operatorname{sat}}}$ is free abelian if for every field $E$, with $F\subseteq E \subseteq K(\Gamma_{\operatorname{sat}})$ and $[E:F]< \infty$, we have

(i) $\nicefrac{{\mathcal{G}}(E)}{(\Gamma_{\operatorname{sat}}\cap {\mathcal{G}}(E))}$ is free abelian, and

(ii) the torsion group of $\nicefrac{{\mathcal{G}}(K(\Gamma_{\operatorname{sat}}))}{{\mathcal{G}}(E) + \Gamma_{\operatorname{sat}}}$ has finite exponent.

This is mainly an application of a classification result of Pontryagin. The proof follows very closely the proofs of [@Ma72], Lemma 1, and [@GHP], Proposition 2.3.

By a theorem of Pontryagin, cf. [@EM], Theorem VI.2.3, an abelian group $G$ is free abelian, if every finite subset of $G$ is contained in a free abelian subgroup $H\subseteq G$ such that $\nicefrac{G}{H}$ is torsion free.

Therefore, let $S=\{[\alpha_1],\ldots,[\alpha_s]\}\subseteq \nicefrac{{\mathcal{G}}(K(\Gamma_{\operatorname{sat}}))}{\Gamma_{\operatorname{sat}}}$ and set $E:=K({\mathcal{G}}_{\operatorname{tors}},\alpha_1,\ldots,\alpha_s)$. Obviously $E$ is a finite extension of $K({\mathcal{G}}_{\operatorname{tors}})$ and it is $$S \subseteq \nicefrac{{\mathcal{G}}(E)+\Gamma_{\operatorname{sat}}}{\Gamma_{\operatorname{sat}}}\cong \nicefrac{{\mathcal{G}}(E)}{{\mathcal{G}}(E)\cap \Gamma_{\operatorname{sat}}}.$$ Let $m\in\mathbb{N}$ be the exponent of the torsion subgroup of $$\nicefrac{\left( \nicefrac{{\mathcal{G}}(K(\Gamma_{\operatorname{sat}}))}{\Gamma_{\operatorname{sat}}} \right)}{\left( \nicefrac{{\mathcal{G}}(E)+\Gamma_{\operatorname{sat}}}{\Gamma_{\operatorname{sat}}}  \right)}\cong \nicefrac{{\mathcal{G}}(K(\Gamma_{\operatorname{sat}}))}{{\mathcal{G}}(E)+\Gamma_{\operatorname{sat}}}.$$ Note, that this exponent is indeed an element of $\mathbb{N}$, by assumption (ii) of the lemma. Now define $$H:=\left\{[\alpha]\in \nicefrac{{\mathcal{G}}(K(\Gamma_{\operatorname{sat}}))}{\Gamma_{\operatorname{sat}}} \vert m\cdot [\alpha] \in \nicefrac{{\mathcal{G}}(E)+\Gamma_{\operatorname{sat}}}{\Gamma_{\operatorname{sat}}} \right\}.$$ By assumption (i) the group $\nicefrac{{\mathcal{G}}(E)+\Gamma_{\operatorname{sat}}}{\Gamma_{\operatorname{sat}}}\cong \nicefrac{{\mathcal{G}}(E)}{{\mathcal{G}}(E)\cap \Gamma_{\operatorname{sat}}}$ is free abelian, and hence $H$ is free abelian. Moreover, by construction the quotient of $\nicefrac{{\mathcal{G}}(K(\Gamma_{\operatorname{sat}}))}{\Gamma_{\operatorname{sat}}}$ by $H$ is torsion free. Hence, $S \subseteq H$ and $H$ satisfies the hypothesis of Pontryagins theorem. It follows that under the assumptions (i) and (ii) $\nicefrac{{\mathcal{G}}(K(\Gamma_{\operatorname{sat}}))}{\Gamma_{\operatorname{sat}}}$ is free abelian.

As stated in the introduction, we will use the result from Bays, Hart, and Pillay that $\nicefrac{{\mathcal{G}}(K({\mathcal{G}}_{\operatorname{tors}}))}{{\mathcal{G}}_{\operatorname{tors}}}$ is free abelian. The case ${\mathcal{G}}=\mathbb{G}_m$ is originally due to Iwasawa [@Iw53], and the case ${\mathcal{G}}=A$ is originally due to Larsen [@La05]. We will sketch a proof of this result.

\[thm:BHP\] Let ${\mathcal{G}}$ and $K$ be as above and let $K'/K$ be finite. Then

(i) $\nicefrac{{\mathcal{G}}(K')}{{\mathcal{G}}(K')\cap{\mathcal{G}}_{\operatorname{tors}}}$ is free abelian, and

(ii) the exponent of the torsion group of $\nicefrac{{\mathcal{G}}(K'({\mathcal{G}}_{\operatorname{tors}}))}{{\mathcal{G}}(K')+{\mathcal{G}}_{\operatorname{tors}}}$ is finite.

In particular, $\nicefrac{{\mathcal{G}}(K({\mathcal{G}}_{\operatorname{tors}}))}{{\mathcal{G}}_{\operatorname{tors}}}$ is a free abelian group.

In this proof we use the language of continuous group cohomology. The field $K'$ is a number field. Hence the norm ${\lVert\cdot\rVert}_h$ induced by the canonical height of ${\mathcal{G}}$ is discrete on $\nicefrac{{\mathcal{G}}(K')}{{\mathcal{G}}(K')\cap{\mathcal{G}}_{\operatorname{tors}}}$. Statement (i) follows from Proposition \[LSZ\]. In order to prove (ii), let $\alpha \in {\mathcal{G}}(K'({\mathcal{G}}_{\operatorname{tors}}))$ be such that $n\cdot \alpha \in {\mathcal{G}}(K')+{\mathcal{G}}_{\operatorname{tors}}$ for some $n \in \mathbb{N}$. Then, the map $\tau \mapsto \tau(\alpha)-\alpha$ represents an element in $H^1(\operatorname{Gal}(K'({\mathcal{G}}_{\operatorname{tors}})/K'),{\mathcal{G}}[n])$. By Lemma A.3 from [@BHP], there is a constant $c$ only depending on ${\mathcal{G}}$ and $K'$ such that $\tau \mapsto c\cdot(\tau(\alpha)-\alpha)$ is equivalent to the zero map in $H^1(\operatorname{Gal}(K'({\mathcal{G}}_{\operatorname{tors}})/K'),{\mathcal{G}}[n])$. Hence, there is an element $P\in{\mathcal{G}}[n]$ such that $$c\cdot(\tau(\alpha)-\alpha)=\tau(P)-P \quad \forall ~ \tau \in \operatorname{Gal}(K'({\mathcal{G}}_{\operatorname{tors}})/K').$$ It follows $$\tau(c\cdot \alpha - P) = c\cdot \alpha - P \quad \forall ~ \tau \in \operatorname{Gal}(K'({\mathcal{G}}_{\operatorname{tors}})/K'),$$ and hence $c\cdot \alpha -P \in {\mathcal{G}}(K')$, respectively $c\cdot \alpha \in {\mathcal{G}}(K') + {\mathcal{G}}_{\operatorname{tors}}$. This means that the order of the residue class of $\alpha \in \nicefrac{{\mathcal{G}}(K'({\mathcal{G}}_{\operatorname{tors}}))}{{\mathcal{G}}_{\operatorname{tors}}}$ divides $c$, which proves statement (ii).

If we apply Lemma \[freemodgamma\] with $\Gamma=\{0\}$ and $F=K$, it follows from (i) and (ii) that $\nicefrac{{\mathcal{G}}(K({\mathcal{G}}_{\operatorname{tors}}))}{{\mathcal{G}}_{\operatorname{tors}}}$ is free abelian.

\[prop:firstclaim\] Let $K'/K$ be finite and let $\Gamma$ be a finite rank subgroup of ${\mathcal{G}}(\overline{\mathbb{Q}})$. For any field $K \subseteq E \subseteq K'({\mathcal{G}}_{\operatorname{tors}})$, the group $\nicefrac{{\mathcal{G}}(E)}{{\mathcal{G}}(E)\cap \Gamma_{\operatorname{sat}}}$ is free abelian.

We have just seen that $\nicefrac{{\mathcal{G}}(K'({\mathcal{G}}_{\operatorname{tors}}))}{{\mathcal{G}}_{\operatorname{tors}}}$ is free abelian. Therefore $\nicefrac{{\mathcal{G}}(E)}{{\mathcal{G}}(E)_{\operatorname{tors}}}$ is, as a subgroup, free abelian. Hence by Proposition \[LSZ\] there is a discrete norm ${\lVert.\rVert}$ on $\nicefrac{{\mathcal{G}}(E)}{{\mathcal{G}}(E)_{\operatorname{tors}}}$.

Set $\tilde{\Gamma}=\Gamma_{\operatorname{sat}}\cap {\mathcal{G}}(E)$ which is a finite rank subgroup, since $\operatorname{End}({\mathcal{G}})$ and $\Gamma$ are of finite rank. The discrete norm ${\lVert.\rVert}$ extends uniquely to a norm on $\nicefrac{{\mathcal{G}}(E)_{\operatorname{div}}}{{\mathcal{G}}_{\operatorname{tors}}}$. Hence we can apply Lemma \[Bogomolov\] to deduce the existence of a constant $c > 0$ such that $$\label{eq:firstpart}
 {\lVert\alpha\rVert}_{\tilde{\Gamma}} = \inf \{{\lVert\alpha+\gamma\rVert} \vert \gamma\in \tilde{\Gamma}_{\operatorname{div}} \} \geq c \text{ for all } \alpha \in {\mathcal{G}}(E)\setminus \tilde{\Gamma}_{\operatorname{div}}.$$ By Lemma \[heightisnorm\] we already know that ${\lVert.\rVert}_{\tilde{\Gamma}}$ is a seminorm on $\nicefrac{{\mathcal{G}}(E)}{\tilde{\Gamma}}$. Therefore tells us that ${\lVert.\rVert}_{\tilde{\Gamma}}$ is actually a discrete norm. If we apply Proposition \[LSZ\] once more, we achieve that $\nicefrac{{\mathcal{G}}(E)}{\tilde{\Gamma}}$ is free abelian.

Proof of Theorem \[thm:freeab\] for the linear torus
====================================================

In this subsection we work with ${\mathcal{G}}=\mathbb{G}_m$; i.e. we work in the multiplicative group of a field. Recall that we denote by $\mu$ the set of all roots of unity.

\[prop:gmtorsionfree\] Let $\Gamma=\langle \gamma_1,\ldots,\gamma_r \rangle$ be a subgroup of $\overline{\mathbb{Q}}^*$, and let $K$ be a number field. We set $L=K(\Gamma_{\operatorname{sat}})$ and let $E\subseteq L$ be a finite extension of $K(\mu,\gamma_1,\ldots,\gamma_r)$. Then the group $\nicefrac{L^*}{E^* \Gamma_{\operatorname{sat}}}$ is torsion free.

Let $[\alpha]\in \nicefrac{L^*}{E^* \Gamma_{\operatorname{sat}}}$ be a torsion point. Then there exists a natural number $n$ with $\alpha^n \in E$.

Since $\alpha$ is in $L=K(\Gamma_{\operatorname{sat}})$, it is $E(\alpha)\subseteq E(\gamma_1^{\nicefrac{1}{m_1}},\dots,\gamma_r^{\nicefrac{1}{m_r}})$ for some $m_1,\dots,m_r \in \mathbb{N}$. We set $E_0 = E$ and define for every $i\in\{1,\dots,r\}$ the field $$E_i = E(\gamma_1^{\nicefrac{1}{m_1}},\dots,\gamma_i^{\nicefrac{1}{m_i}}).$$ Every extension $E_i / E_{i-1}$ in the chain $$E=E_0 \subseteq E_1 \subseteq E_2 \subseteq \cdots \subseteq E_r$$ is cyclic of some order $k_i \mid m_i$. Note, that $\gamma_i \in E$, for all $i\in \{1,\dots,r\}$. By the classic theory of cyclic extensions (cf. [@La02], Chapter VI), every intermediate field of $E_{i}=E_{i-1}(\gamma_i^{\nicefrac{1}{m_i}})/E_{i-1}$ is given by $E_{i-1}(\gamma_i^{\nicefrac{d}{m_i}})$ for some $d\in\mathbb{N}$. Hence, $E_{r-1}(\alpha)=E_{r-1}(\gamma_r^{\nicefrac{d_r}{m_r}})$. Assume, that the degree of $\alpha$ over $E_{r-1}$ is $n_r$ and that $\sigma$ is a generator of $\operatorname{Gal}(E_{r-1}(\alpha)/E_{r-1})$. Then, since $\alpha^n \in E \subseteq E_{r-1}$, we have $$\sigma(\gamma_r^{\nicefrac{d_r}{m_r}})=\zeta_{n_r}\gamma_r^{\nicefrac{d_r}{m_r}} \quad \text{ and } \quad \sigma(\alpha)=\zeta_{n_r}^{l_r} \alpha,$$ where $\zeta_{n_r}$ is a primitive $n_r$-th root of unity and $l_r \in \mathbb{N}$. We can conclude that $\sigma$, and hence $\operatorname{Gal}(E_{r-1}(\alpha)/E_{r-1})$, acts trivial on the element $\nicefrac{\alpha}{\gamma_r^{\nicefrac{d_r l_r}{m_r}}}$. It follows $$\frac{\alpha}{\gamma_r^{\nicefrac{d_r l_r}{m_r}}} \in E_{r-1} \quad \text{ and } \quad \left(\frac{\alpha}{\gamma_r^{\nicefrac{d_r l_r}{m_r}}}\right)^{n m_r} \in E.$$ Thus we can repeat this argument with $n$ replaced by $n m_r$, and $\alpha$ replaced by $\nicefrac{\alpha}{\gamma_r^{\nicefrac{d_r l_r}{m_r}}}$. Induction yields $$\frac{\alpha}{\prod_{i=1}^{r}\gamma_i^{\nicefrac{d_i l_i}{m_i}}} \in E_0^*$$ which is equivalent to $\alpha \in E^* \Gamma_{\operatorname{sat}}$. Hence the residue class of $\alpha$ in the group $\nicefrac{L^*}{E^* \Gamma_{\operatorname{sat}}}$ is trivial, meaning that the group is torsion free.

Now let $L$ be a finite extension of $K(\Gamma_{\operatorname{sat}})$; say $L=K'(\Gamma_{\operatorname{sat}})$ for a finite extension $K'/K$. Set $F=K'(\gamma_{1},\ldots,\gamma_{r},\mu)$ and let $E/F$ be any finite extension such that $E\subseteq L$. By Proposition \[prop:firstclaim\] we know that $\nicefrac{E^*}{E^*\cap \Gamma_{\operatorname{div}}}$ is free abelian and we have just seen that $\nicefrac{L^*}{E^* \Gamma_{\operatorname{div}}}$ is torsion free. Hence, the assumptions from Lemma \[freemodgamma\] are met, which proves Theorem \[thm:freeab\] for ${\mathcal{G}}=\mathbb{G}_m$.

Preliminaries on elliptic curves
================================

In this section $K$ denotes a number field and we fix an elliptic curve $A$ defined over $K$. We also fix a finite rank subgroup $\Gamma \subseteq A(\overline{\mathbb{Q}})$. There are $\mathbb{Z}$-linearly independent elements $\gamma_1,\ldots,\gamma_s$ such that $\Gamma_{\operatorname{div}} = \langle \gamma_1,\ldots,\gamma_s\rangle_{\operatorname{div}}$. If $\gamma_1,\ldots,\gamma_s$ are $\operatorname{End}(A)$-linearly dependent (which can only occur if $A$ has complex multiplication), then there are $\phi_1,\ldots,\phi_s \in \operatorname{End}(A)$ not all constantly zero such that $\phi_1(\gamma_1)+\ldots+\phi_s(\gamma_s) = 0$. We may assume that $\phi_s\neq 0$, and that $\hat{\phi_s}$ is the dual of $\phi_s$. Then $$-(\hat{\phi_s}\circ\phi_1(\gamma_1)+\ldots+\hat{\phi_s}\circ\phi_{s-1}(\gamma_{s-1})) = \hat{\phi_s}\circ\phi_s(\gamma_s)=\deg(\phi_s)\gamma_s.$$ We find that $\gamma_s \in (\operatorname{End}(A)\cdot \gamma_1+\ldots+\operatorname{End}(A)\cdot \gamma_{s-1})_{\operatorname{div}}$, and hence in $\Gamma_{\operatorname{sat}}=(\operatorname{End}(A)\cdot \gamma_1+\ldots+\operatorname{End}(A)\cdot \gamma_{s-1})_{\operatorname{div}}$. Therefore, after possibly shrinking the set of generators $\gamma_1,\ldots,\gamma_s$, we may assume $$\begin{aligned}
\label{eq:Endlinin}
\Gamma_{\operatorname{sat}} &=(\operatorname{End}(A)\cdot \Gamma)_{\operatorname{div}}=(\operatorname{End}(A)\cdot \gamma_1 + \ldots + \operatorname{End}(A)\cdot \gamma_r)_{\operatorname{div}} \\  &\text{ with } \gamma_1,\ldots,\gamma_r ~ \operatorname{End}(A)\text{-linearly independent}. \nonumber\end{aligned}$$ The group $\nicefrac{A(K(A_{\operatorname{tors}}))}{A_{\operatorname{tors}}}$ is free abelian (see Theorem \[thm:BHP\]). Hence, after replacing $\gamma_i$ by some division point of $\gamma_i$, we may assume that for all prime numbers $\ell$ it is $$\label{eq:divisionprop}
\gamma_i \in A(K(A_{\operatorname{tors}})) \text{ and } \frac{1}{\ell}\gamma_i \notin A(K(A_{\operatorname{tors}})) \quad \forall ~ i\in\{1,\ldots,r\}.$$ Moreover we assume, that $\gamma_1,\ldots,\gamma_r\in A(K)_{\operatorname{div}}$ and $\operatorname{End}(A)$ is defined over $K$, which is always possible after replacing $K$ by a finite extension. In particular, we assume $$\label{eq:alloverK}
\Gamma':=(\operatorname{End}(A)\cdot \gamma_1 + \ldots + \operatorname{End}(A)\cdot \gamma_r) \subseteq A(K)_{\operatorname{div}}.$$

For $n\in\mathbb{N}$ denote the group of $n$-torsion points of $A$ by $A[n]$. We start with collecting some basic facts:

-   For any $n\in\mathbb{N}$ it is $A[n]\cong \left(\nicefrac{\mathbb{Z}}{n\mathbb{Z}}\right)^{2}$.

-   Let $F$ be a subfield of $\overline{\mathbb{Q}}$, with $A_{\operatorname{tors}}\subseteq A(F)$, and let $\gamma \in A(F)$. Then, $F(\frac{1}{n}\gamma)/F$ is a Galois extension and independent on the choice of the $n$-th division point of $\gamma$. Moreover, the map $$\operatorname{Gal}(F(\frac{1}{n}\gamma)/F)\longrightarrow A[n]\quad ; \quad \sigma \mapsto \sigma(\frac{1}{n}\gamma)-\frac{1}{n}\gamma$$ is an injective group homomorphism.

-   In the situation above, let $n=k\cdot m$ with $k$ and $m$ coprime. Then $F(\frac{1}{n}\gamma)=F(\frac{1}{k}\gamma,\frac{1}{m}\gamma)$.

We will use these facts freely for the remainder of this paper. If $\ell$ is a prime, $n\in\mathbb{N}$ and $G$ is any subgroup of $A(\overline{\mathbb{Q}})$, then we set $$\frac{1}{\ell^n} G=\{\gamma \in {\mathcal{G}}(\overline{\mathbb{Q}}) \vert \ell^n \gamma \in G\} \quad \text{ and } \quad \frac{1}{\ell^{\infty}} G= \bigcup_{n\in\mathbb{N}} \frac{1}{\ell^n} G$$ Moreover, we define $$G_{\operatorname{tors}}=\operatorname{Gal}(K(A_{\operatorname{tors}})/K).$$

Throughout this section we use the notation from above. In particular, we assume that the assumptions , and are met.

\[lem:intersection\] For any prime number $\ell$ and any $n\in\mathbb{N}$ it is $$\Gamma' + A_{\operatorname{tors}} = \left( \frac{1}{\ell^{n}}\Gamma' \cap A(K(A_{\operatorname{tors}}))\right) + A_{\operatorname{tors}}.$$

We set $\widetilde{\Gamma_n}= \left( \frac{1}{\ell^{n}}\Gamma' \cap A(K(A_{\operatorname{tors}}))\right)$. By , $\nicefrac{\widetilde{\Gamma_n}+A_{\operatorname{tors}}}{A_{\operatorname{tors}}}$ is a free $\operatorname{End}(A)$-module of rank $r$. Moreover, it is $$\nicefrac{\Gamma'+A_{\operatorname{tors}}}{A_{\operatorname{tors}}}\subseteq \nicefrac{\widetilde{\Gamma_n}+A_{\operatorname{tors}}}{A_{\operatorname{tors}}} \subseteq \nicefrac{\frac{1}{\ell^n}\Gamma'+A_{\operatorname{tors}}}{A_{\operatorname{tors}}}.$$ Now, implies $\nicefrac{\Gamma'+A_{\operatorname{tors}}}{A_{\operatorname{tors}}} = \nicefrac{\widetilde{\Gamma_n}+A_{\operatorname{tors}}}{A_{\operatorname{tors}}}$.

\[lem:fellinjective\] For any prime number $\ell$ and any $n\in\mathbb{N}$ the map $$f_{\ell^n} : \nicefrac{\Gamma'}{\ell^n \Gamma'}\longrightarrow \operatorname{Hom}(\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}})),A[\ell^n]) \quad ; \quad [\gamma] \mapsto \left[ \varphi_\gamma : \sigma \mapsto \sigma(\frac{1}{\ell^n}\gamma) - \frac{1}{\ell^n}\gamma\right]$$ is an injective group-homomorphism.

Let $\gamma \in \Gamma'$ be arbitrary. From the facts collected above, it is clear that $\varphi_\gamma$ is indeed a well-defined group homomorphism. Moreover, for any $\gamma' \in \Gamma'$, it is $$\varphi_{\gamma+\ell^n\gamma'}(\sigma) = \sigma(\frac{1}{\ell^n}\gamma + \gamma')-(\frac{1}{\ell^n}\gamma + \gamma') =\sigma(\frac{1}{\ell^n}\gamma) + \sigma(\gamma') -\gamma' -\frac{1}{\ell^n}\gamma =\sigma(\frac{1}{\ell^n}\gamma)-\frac{1}{\ell^n}\gamma =\varphi_{\gamma}(\sigma)$$ for all $\sigma \in \operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}}))$. Hence, the map $f_{\ell^n}$ is well-defined and visibly an homomorphism. Moreover, $$\begin{aligned}
[\gamma] \in \ker(f_{\ell^n}) ~ & \Longrightarrow ~  \sigma(\frac{1}{\ell^n}\gamma)=\frac{1}{\ell^n}\gamma  \quad \forall ~\sigma \in \operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}})) \\
& \Longrightarrow ~ \frac{1}{\ell^n}\gamma \in \frac{1}{\ell^{n}}\Gamma'\cap A(K(A_{\operatorname{tors}})) \overset{\ref{lem:intersection}}{\subseteq} \Gamma' + A_{\operatorname{tors}}  $$ By and , the group $\Gamma'$ is torsion-free. Hence, if $\frac{1}{\ell^n}\gamma=\gamma'+T$ for some $\gamma'\in\Gamma'$ and some $T\in A_{\operatorname{tors}}$, then $T\in \frac{1}{\ell^n}\Gamma' \cap A_{\operatorname{tors}}=A[\ell^n]$. This implies $\gamma = \ell^n \gamma' + \ell^n T=\ell^n\gamma' \in\ell^n \Gamma'$; i.e. $[\gamma]=[0]$. Hence $f_{\ell^n}$ is injective, which proves the lemma.

\[lem:imageisGstable\] For any $\gamma \in \Gamma'$, the map $\varphi_{\gamma}$ from Lemma \[lem:fellinjective\] is a $G_{\operatorname{tors}}$-module homomorphism.

Let $\tau\in G_{\operatorname{tors}}$ and $\sigma \in \operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}}))$ be arbitrary, and chose an extension of $\tau$ to an element from $\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K)$ which we will again denote by $\tau$. Using the canonical isomorphism $$\nicefrac{\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K)}{\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}}))}\cong G_{\operatorname{tors}},$$ there is a well-defined element $\tau\circ\sigma\circ\tau^{-1}\in\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}}))$. Indeed, the group $\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}}))$ is abelian and hence for any $\tau'\in \operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}}))$ it is $$(\tau\circ\tau')\circ\sigma\circ(\tau\circ\tau')^{-1}=\tau\circ\sigma\circ\tau^{-1}.$$ This conjugation gives $\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}}))$ the structure of a $G_{\operatorname{tors}}$-module. Since $\frac{1}{\ell^n}\gamma\in\Gamma_{\operatorname{sat}}$, assumptions and imply that some multiple of $\frac{1}{\ell^n}\gamma$ is defined over $K$. In particular, $\tau^{-1}(\frac{1}{\ell^n}\gamma)-\frac{1}{\ell^n}\gamma \in A_{\operatorname{tors}}$, and hence $$\sigma\circ\tau^{-1}(\frac{1}{\ell^n}\gamma) - \sigma(\frac{1}{\ell^n}\gamma)=\sigma(\tau^{-1}(\frac{1}{\ell^n}\gamma)-\frac{1}{\ell^n}\gamma)=\tau^{-1}(\frac{1}{\ell^n}\gamma)-\frac{1}{\ell^n}\gamma.$$ Applying $\tau$ on both sides implies $$\tau(\varphi_{\gamma}(\sigma))=\tau(\sigma(\frac{1}{\ell^n}\gamma)-\frac{1}{\ell^n}\gamma)=\tau\circ\sigma\circ\tau^{-1}(\frac{1}{\ell^n}\gamma)-\frac{1}{\ell^n}\gamma = \varphi_{\gamma}(\tau\sigma\tau^{-1}),$$ which proves the lemma.

\[cor:Galois\] The following properties hold true:

(i) $\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}}))$ is isomorphic to $$\{(\varphi_{\gamma_1}(\tau),\ldots,\varphi_{\gamma_r}(\tau)) \vert \tau \in \operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}}))\} \subseteq A[\ell^n]^r$$ as $G_{\operatorname{tors}}$-modules, and

(ii) the maps $\varphi_{\gamma_1},\ldots,\varphi_{\gamma_r}$ are $\nicefrac{\operatorname{End}(A)}{\ell^n\operatorname{End}(A)}$-linearly independent.

The map $\tau \mapsto (\varphi_{\gamma_1}(\tau),\ldots,\varphi_{\gamma_r}(\tau))$ is obviously surjective, and it is a $G_{\operatorname{tors}}$-module homomorphism by Lemma \[lem:imageisGstable\]. An element $\tau$ in the kernel must fix all of the points $\frac{1}{\ell^n}\gamma_1,\ldots,\frac{1}{\ell^n}\gamma_r$. Since $\operatorname{End}(A)$ is defined over $K$, this element fixes all elements in $\frac{1}{\ell^n} \Gamma'$. Hence, such $\tau$ is the identity on $K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')$. Therefore, the kernel is trivial, which concludes the proof of part (i).

Let $a_1\cdot \varphi_{\gamma_1} + \ldots + a_r\cdot \varphi_{\gamma_r}=0$, with $a_1,\ldots,a_r\in\operatorname{End}(A)$. Then $\gamma=a_1\cdot\gamma_1 + \ldots + a_r\cdot \gamma_r$ satisfies $$\begin{gathered}
\varphi_{\gamma}(\sigma) =\sigma(\frac{1}{\ell^n}\gamma)-\frac{1}{\ell^n}\gamma = a_1\cdot \varphi_{\gamma_1}(\sigma) + \ldots + a_r\cdot \varphi_{\gamma_r}(\sigma) =0 \nonumber \\ \text{ for all } \sigma \in \operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}})).\end{gathered}$$ By Lemma \[lem:fellinjective\], it follows $\gamma\in\ell^n\Gamma'$. Since $\gamma_1,\ldots,\gamma_r$ are $\operatorname{End}(A)$-linearly independent, it is $a_i\in \ell^n \cdot \operatorname{End}(A)$ for all $i\in\{1,\ldots,r\}$, proving part (ii).

If $M$ and $N$ are two $G_{\operatorname{tors}}$-modules, then as usual $\operatorname{Hom}_{G_{\operatorname{tors}}}(M,N)$ is the set of all $G_{\operatorname{tors}}$-module homomorphisms from $M$ to $N$. Hence, the combination of Lemmas \[lem:fellinjective\] and \[lem:imageisGstable\] gives $$\label{eq:lowerboundhom}
\ell^{n\cdot r} = \left\vert \nicefrac{\Gamma'}{\ell^n \Gamma'}\right\vert \leq \vert\operatorname{Hom}_{G_{\operatorname{tors}}}(\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^n}\Gamma')/K(A_{\operatorname{tors}})),A[\ell^n])\vert$$

\[lem:restriction\] Assume that $\alpha\in A(K(A_{\operatorname{tors}},\Gamma_{\operatorname{sat}}))$ satisfies $\ell^n \alpha \in A(K(A_{\operatorname{tors}}))$. Then $\alpha \in A(K(A_{\operatorname{tors}},\frac{1}{\ell^{\infty}}\Gamma'))$.

For any $k\in\mathbb{N}$ it is (since $\operatorname{End}(A)$ is defined over $K$) $$\label{eq:generator}
K(A_{\operatorname{tors}},\frac{1}{\ell^k}\Gamma')=K(A_{\operatorname{tors}},\frac{1}{\ell^k}\gamma_1,\ldots,\frac{1}{\ell^k}\gamma_k).$$ Let $n_1,\ldots,n_r \in \mathbb{N}$ be such that $$\alpha\in A(K(A_{\operatorname{tors}},\frac{1}{n_1}\gamma_1 ,\ldots ,\frac{1}{n_r}\gamma_r)).$$ For all $i \in \{1,\ldots,r\}$ we write $n_i=\ell^{e_i} m_i$ with $\ell\nmid m_i$, and set $$K_i:=K(A_{\operatorname{tors}},\frac{1}{n_1}\gamma_1,\ldots,\frac{1}{n_{i-1}}\gamma_{i-1},\frac{1}{\ell^{e_i}}\gamma_i,\frac{1}{n_{i+1}}\gamma_{i+1},\ldots,\frac{1}{n_r}\gamma_r).$$ Then

-   $G_i:=\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{n_1}\gamma_1,\ldots ,\frac{1}{n_r}\gamma_r)/ K_i) \hookrightarrow A[m_i]$, and

-   $G_{i,\alpha}:= \operatorname{Gal}(K_i(\alpha)/ K_i) \hookrightarrow A[\ell^{n}]$, and

-   $G_{i,\alpha}$ is isomorphic to a quotient of $G_i$.

Since $\gcd(\vert A[\ell^{e_i}] \vert , \vert A[m_i]\vert)=1$, this is only possible if $G_{i,\alpha}$ is trivial and hence $\alpha \in A(K_i)$. Therefore, we can assume $m_i=1$. This is true for all $i\in\{1,\ldots,r\}$ and hence, we have $$\alpha\in A(K(A_{\operatorname{tors}},\frac{1}{\ell^{e_1}}\gamma_1,\ldots,\frac{1}{\ell^{e_r}}\gamma_r))\subseteq A(K(A_{\operatorname{tors}},\frac{1}{\ell^{\infty}}\Gamma')).$$

Proof of Theorem \[thm:freeab\] for elliptic curves with complex multiplication
===============================================================================

From now on let $A$ be an elliptic curve defined over a number field $K$ with complex multiplication. Let $\mathcal{O}$ be the order in an imaginary quadratic field $\kappa$ such that $\operatorname{End}(A)\cong \mathcal{O}$. By enlarging $K$ we may assume $\kappa\subseteq K$. For any prime number $\ell$ and any $m\in\mathbb{N}$ it is $A[\ell^m]\cong \nicefrac{\mathcal{O}}{\ell^m \mathcal{O}}$ and $\operatorname{Gal}(K(A[\ell^m])/K)$ acts on $A[\ell^m]\cong \nicefrac{\mathcal{O}}{\ell^m \mathcal{O}}$ as multiplication by units in $\nicefrac{\mathcal{O}}{\ell^m \mathcal{O}}$. Therefore, we can regard $\operatorname{Gal}(K(A[\ell^m])/K)$ as a subgroup of $\left(\nicefrac{\mathcal{O}}{\ell^m \mathcal{O}}\right)^*$.

Up to this restriction, the fields generated by torsion points are almost as large as possible. Namely, $$\label{eq:serreCM}
(\left(\nicefrac{\mathcal{O}}{\ell^m \mathcal{O}}\right)^*:\operatorname{Gal}(K(A[\ell^m])/K))\leq M,$$ for some absolute constant $M$ only depending on $A$ and $K$ (cf. [@Se], Section 4.5).

Let $f$ be the conductor of $\mathcal{O}$ and define for all primes $\ell$ the positive integer $$m_{\ell} \text{ minimal such that } \ell^{m_{\ell}} > 4 \ell^{\operatorname{ord}_\ell (f)}  M.$$ It is obvious that $m_\ell=1$ for all but finitely many primes $\ell$.

\[lem:indexCM\] We use the notation from above. Then it is $$(\nicefrac{\mathcal{O}}{\ell^m \mathcal{O}}:\nicefrac{\mathbb{Z}}{\ell^m \mathbb{Z}}[\operatorname{Gal}(K(A[\ell^m])/K)]) \leq 4 \ell^{\operatorname{ord}_\ell (f)} M < \ell^{m_l}.$$ Here we identify $\operatorname{Gal}(K(A[\ell^m])/K)$ with a subgroup of $\left(\nicefrac{\mathcal{O}}{\ell^m \mathcal{O}}\right)^*$.

We denote by $\mathcal{O}_{\kappa}$ the ring of integers of $\kappa$. Then the map $$\nicefrac{\mathcal{O}}{\ell^m \mathcal{O}} \longrightarrow \nicefrac{\mathcal{O}_{\kappa}}{\ell^m \mathcal{O}_{\kappa}} \quad ; \quad a+\ell^m \mathcal{O} \mapsto a +\ell^m\mathcal{O}_{\kappa}$$ is a $\ell^{\operatorname{ord}_\ell (f)}$-to-$1$ ring-homomorphism. In particular we have $$\label{eq:CM1}
\vert \left(\nicefrac{\mathcal{O}}{\ell^m \mathcal{O}}\right)^* \vert \geq \frac{\vert \left(\nicefrac{\mathcal{O}_{\kappa}}{\ell^m \mathcal{O}_{\kappa}}\right)^* \vert}{\ell^{\operatorname{ord}_\ell (f)}}.$$ This last quantity can be explicitly calculated and it is $$\label{eq:CM2}
\vert \left(\nicefrac{\mathcal{O}_{\kappa}}{\ell^m \mathcal{O}_{\kappa}}\right)^* \vert = \ell^{2m} (1-\ell^{-1}) (1-\left(\frac{\operatorname{disc}_{\kappa}}{\ell}\right) \ell^{-1}),$$ where $\operatorname{disc}_{\kappa}$ is the discriminant of $\kappa/\mathbb{Q}$ and $\left(\frac{\operatorname{disc}_{\kappa}}{\ell}\right)$ is the Legendre symbol (this follows from solving exercise 7.29 from [@Cox]). Using , and the definition of $M$ we get $$\vert \nicefrac{\mathbb{Z}}{\ell^m \mathbb{Z}}[\operatorname{Gal}(K(A[\ell^m])/K)] \vert \geq \vert \operatorname{Gal}(K(A[\ell^m])/K) \vert \geq \frac{\vert\left(\nicefrac{\mathcal{O}}{\ell^m \mathcal{O}}\right)^*\vert}{M} \geq \frac{\ell^{2m} (1-\ell^{-1})^2}{M \ell^{\operatorname{ord}_\ell (f)}}.$$ It immediately follows $$(\nicefrac{\mathcal{O}}{\ell^m \mathcal{O}}:\nicefrac{\mathbb{Z}}{\ell^m \mathbb{Z}}[\operatorname{Gal}(K(A[\ell^m])/K)]) \leq \frac{\ell^{2m} \ell^{\operatorname{ord}_\ell (f)} M}{\ell^{2m} (1-\ell^{-1})^2} \leq 4 \ell^{\operatorname{ord}_\ell (f)} M.$$

In order to prove Theorem \[thm:freeab\] for elliptic curves with complex multiplication, let $\Gamma \subseteq A(\overline{\mathbb{Q}})$ be a subgroup of finite rank.

We have to prove that $\nicefrac{A(L)}{\Gamma_{\operatorname{sat}}}$ is free abelian for any finite extension $L$ of $K(\Gamma_{\operatorname{sat}})$. Since the property of being free abelian is inherited by subgroups, we may enlarge the field $L$ as we please. In particular, we may assume $\gamma_1,\ldots,\gamma_r\in A(L)$.

After replacing our base field $K$ by a finite extension, we can also assume $L=K(\Gamma_{\operatorname{sat}})$. Thanks to Lemma \[freemodgamma\] and Proposition \[prop:firstclaim\] it is enough to prove that the torsion group of $\nicefrac{A(L)}{A(K')+\Gamma_{\operatorname{sat}}}$ has finite exponent for all finite extensions $K'/K$. This is surely the case if the statement is true for $\nicefrac{A(K'L)}{A(K')+\Gamma_{\operatorname{sat}}}$. Hence we have to prove that the torsion group of $$\nicefrac{A(K(\Gamma_{\operatorname{sat}}))}{A(K)+\Gamma_{\operatorname{sat}}}$$ has finite exponent for all sufficiently large number fields $K$ over which $A$ is defined. Hence, we fix a number field $K$, with $A$ is defined over $K$, and such that there are $\gamma_1,\ldots,\gamma_r$ satisfying properties , and (in particular it is $\kappa\subseteq K$).

Let $c$ be the exponent of the torsion group of $\nicefrac{A(K(A_{\operatorname{tors}}))}{A(K)+A_{\operatorname{tors}}}$ (cf. Theorem \[thm:BHP\]).

Let $[\alpha] \in \nicefrac{A(K(\Gamma_{\operatorname{sat}}))}{A(K)+\Gamma_{\operatorname{sat}}}$ be of order $\ell^{m_\ell+ \operatorname{ord}_{\ell}(c)}$, where $\ell$ is a prime number. By possible changing the representative of $[\alpha]$, we can assume that $\ell^{m_\ell+ \operatorname{ord}_{\ell}(c)} \alpha \in A(K)$. By definition of $c$, it is $\ell^{m_{\ell}-1}\alpha \notin A(K(A_{\operatorname{tors}}))$, since otherwise $\ell^{m_{\ell}-1}\alpha$ would be an element of order $\ell^{\operatorname{ord}_{\ell}(c)+1}\nmid c$ in $\nicefrac{A(K(A_{\operatorname{tors}}))}{A(K)+A_{\operatorname{tors}}}$. Hence, $$\ell^m \alpha \in A(K(A_{\operatorname{tors}})) \quad \text{ and } \quad \ell^{m-1}\alpha \notin A(K(A_{\operatorname{tors}})) \text{ for some } m\geq m_{\ell}.$$

If $\gamma_1,\ldots,\gamma_r,\alpha$ would be $\mathcal{O}$-linearly dependent, then – as seen in Section 3 – $\alpha \in \Gamma_{\operatorname{sat}}$ which is not the case. Hence $\gamma_1,\ldots,\gamma_r,\ell^m \alpha$ are $\mathcal{O}$-linearly independent and satisfy , and with $\Gamma$ replaced by $\Gamma_{\alpha}=\langle\Gamma,\alpha\rangle$ and $\Gamma'$ replaced by $\Gamma_{\alpha}'=\mathcal{O}\cdot \langle\gamma_1,\ldots,\gamma_r,\ell^m \alpha\rangle$.

Therefore, Corollary \[cor:Galois\] tells us that $\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^m}\Gamma_{\alpha}')/K(A_{\operatorname{tors}}))$ is a $\nicefrac{\mathbb{Z}}{\ell^m\mathbb{Z}}[\operatorname{Gal}(K(A[\ell^m])/K)]$-submodule of $A[\ell^{m}]^{r+1} \cong \left(\nicefrac{\mathcal{O}}{\ell^{m} \mathcal{O}}\right)^{r+1}$.

Let $W\subseteq \left(\nicefrac{\mathcal{O}}{\ell^{m} \mathcal{O}}\right)^{r+1}$ be the $\mathcal{O}$-module generated by $\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^{m}}\Gamma_{\alpha}')/K(A_{\operatorname{tors}}))$. Hence, any element in $W$ is the sum of elements of the form $a \sigma$ with $\sigma \in \operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^{m}}\Gamma_{\alpha}')/K(A_{\operatorname{tors}}))$ and $a \in \mathcal{O}$. By Lemma \[lem:indexCM\] we find that $$\label{eq:Wexponent}
\text{the exponent of } \nicefrac{W}{\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^{m}}\Gamma_{\alpha}')/K(A_{\operatorname{tors}}))} \text{ is at most } 4M\ell^{ord_{\ell}(f)}.$$ Assume that $W\neq \left(\nicefrac{\mathcal{O}}{\ell^{m} \mathcal{O}}\right)^{r+1}$, then there are $a_1,\ldots,a_{r+1} \in \mathcal{O}$, not all congruent to zero modulo $\ell^{m}\mathcal{O}$, such that $$a_1 w_1 + \ldots + a_{r+1} w_{r+1} \equiv 0 \mod{\ell^{m}\mathcal{O}} \text{ for all } (w_1,\ldots,w_{r+1})\in W$$ (see Lemma 3 in V§5 of [@La]). In particular, for all $\sigma \in \operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^{m}}\Gamma_{\alpha}')/K(A_{\operatorname{tors}}))$ it is $$a_1 \varphi_{\gamma_1}(\sigma) + \ldots + a_r \varphi_{\gamma_r}(\sigma) + a_{r+1} \varphi_{\ell^{m}\alpha}(\sigma) \equiv 0 \mod{\ell^{m}\mathcal{O}},$$ where $\varphi_{\gamma}(\sigma)=\sigma(\frac{1}{\ell^{m}}\gamma)-\frac{1}{\ell^{m}}\gamma$. This is a contradiction, since the maps $\varphi_{\gamma_1},\ldots,\varphi_{\gamma_r},\varphi_{\ell^{m}\alpha}$ are $\nicefrac{\mathcal{O}}{\ell^{m}\mathcal{O}}$-linearly independent by Corollary \[cor:Galois\].

We conclude, that $W=\left(\nicefrac{\mathcal{O}}{\ell^{m} \mathcal{O}}\right)^{r+1}$.

If it were not possible to embed $A[\ell]^{r+1}\cong\left(\nicefrac{\mathcal{O}}{\ell\mathcal{O}}\right)^{r+1}$ into $\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^{m}}\Gamma_{\alpha}')/K(A_{\operatorname{tors}}))$, then the exponent of $$\nicefrac{A[\ell^{m}]^{r+1}}{\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^{m}}\Gamma_{\alpha}')/K(A_{\operatorname{tors}}))} \cong \nicefrac{W}{\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^{m}}\Gamma_{\alpha}')/K(A_{\operatorname{tors}}))}$$ would be equal to $\ell^{m}$. Then implies that $\ell^m \leq 4M\ell^{ord_{\ell}(f)} \overset{\eqref{eq:serreCM}}{<} \ell^{m_\ell}$ in contradiction to the fact $m\geq m_{\ell}$. Therefore, $A[\ell]^{r+1}$ is isomorphic to a subgroup of $\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^{m_\ell}}\Gamma_{\alpha}')/K(A_{\operatorname{tors}}))$.

By the classification of finite abelian groups, there is a surjective group homomorphism $$\label{eq:surjectionm}
\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^{m}}\Gamma_{\alpha}')/K(A_{\operatorname{tors}})) \twoheadrightarrow A[\ell]^{r+1}.$$ On the other hand, $\alpha$ is in $A(K(\Gamma_{\operatorname{sat}}))=A(K(\mathcal{O}\cdot \langle \gamma_1,\ldots,\gamma_r\rangle)_{\operatorname{div}})$ and hence there is an $N\in\mathbb{N}$ such that $\alpha \in A(K(A_{\operatorname{tors}},\frac{1}{\ell^N} (\mathcal{O}\cdot \langle \gamma_1,\ldots,\gamma_r\rangle))$ by Lemma \[lem:restriction\]. Therefore, there is a surjective group homomorphism $$\begin{aligned}
\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^N} (\mathcal{O}\cdot \langle \gamma_1,\ldots,\gamma_r\rangle))/K(A_{\operatorname{tors}})) &\twoheadrightarrow \operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^m} (\mathcal{O}\cdot \langle \gamma_1,\ldots,\gamma_r,\ell^m\alpha\rangle))/K(A_{\operatorname{tors}})) \\  &\overset{\eqref{eq:surjectionm}}{\twoheadrightarrow} A[\ell]^{r+1}.\end{aligned}$$ This is a contradiction to the fact that $\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^N} (\mathcal{O}\cdot \langle \gamma_1,\ldots,\gamma_r\rangle))/K(A_{\operatorname{tors}}))$ is isomorphic to a subgroup of $A[\ell^N]^r$. We conclude that there is no point of order $\ell^{m_{\ell}+\operatorname{ord}_{\ell}(c)}$ in $\nicefrac{A(K(\Gamma_{\operatorname{sat}}))}{A(K)+\Gamma_{\operatorname{sat}}}$. It follows that the exponent of the torsion subgroup of this group is a divisor of $$c\cdot \prod_{\ell \text{ prime}} \ell^{m_{\ell}-1}$$ and hence finite, since $m_{\ell}=1$ for all but finitely many primes $\ell$.

Fields generated by finite rank subgroups of elliptic curves without complex multiplication
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From now on let $A/K$ be an elliptic curve without complex multiplication and let $\ell$ be a prime number. We will use Serre’s famous open image theorem [@Se], Section 4.4. This implies that there exists an absolute constant $M$ only depending on $A$ and $K$, such that $$\label{eq:serre}
(GL_2(\nicefrac{\mathbb{Z}}{\ell^n\mathbb{Z}}):\operatorname{Gal}(K(A[\ell^n])/K))\leq M.$$ With this constant we define $$\label{eq:ml}
m_\ell\in \mathbb{N} \text{ minimal such that } M! < \ell^{m_\ell}.$$ Obviously, for all but finitely many prime numbers it is $m_\ell =1$.

\[lem:fullorbit\] Let $m\geq m_\ell$ and $P\in A[\ell^{m}]$ be of exact order $\ell^{m}$. Then $\nicefrac{A[\ell^m]}{\mathbb{Z}[G_{\operatorname{tors}}]\cdot P}$ is cyclic of order dividing $\ell^{m_\ell-1}$. In particular, $A[\ell]$ is a simple $\mathbb{Z}[G_{\operatorname{tors}}]$-module for all but finitely many prime numbers $\ell$.

Let $P$ be of order $\ell^m$ and let $P' \in A[\ell^m]$ be such that $\{P,P'\}$ is a $\nicefrac{\mathbb{Z}}{\ell^m \mathbb{Z}}$-basis of $A[\ell^m]$. We represent the elements of $GL_2(\nicefrac{\mathbb{Z}}{\ell^m \mathbb{Z}})$ in this basis and regard $\operatorname{Gal}(K(A[\ell^m])/K)$ as a subset of $GL_2(\nicefrac{\mathbb{Z}}{\ell^m \mathbb{Z}})$.

Let $N\leq M!$ be the index of the normal core of $\operatorname{Gal}(K(A[\ell^m])/K)$ in $GL_2(\nicefrac{\mathbb{Z}}{\ell^m \mathbb{Z}})$. Then $$\sigma=\begin{pmatrix}
1 & 0 \\ 1 & 1
\end{pmatrix}^N = \begin{pmatrix}
1 & 0 \\ N & 1
\end{pmatrix} \in \operatorname{Gal}(K(A[\ell^m])/K).$$ Now $\sigma(P)-P=N\cdot P' \in \mathbb{Z}[G_{\operatorname{tors}}]\cdot P$. Let $\ell^n$ be the largest $\ell$-power dividing $N$, then $\nicefrac{N}{\ell^n}\in\left(\nicefrac{\mathbb{Z}}{\ell^m\mathbb{Z}}\right)^*$ and hence $\ell^n \cdot P' \in \mathbb{Z}[G_{\operatorname{tors}}]\cdot P$. Therefore, $\nicefrac{A[\ell^m]}{\mathbb{Z}[G_{\operatorname{tors}}]\cdot P}$ is a cyclic group of order dividing $\ell^n \leq \ell^{m_\ell -1}$. This proves the first statement of the lemma. The second statement follows, since for all but finitely many primes $\ell$ it is $m_{\ell}=1$.

\[cor:homsmall\] We use the notation from the previous lemma. Then $\vert \operatorname{Hom}_{G_{\operatorname{tors}}}(A[\ell^m],A[\ell^m])\vert \leq \ell^m \cdot \ell^{3(m_{\ell} -1)}$. In particular, $\operatorname{Hom}_{G_{\operatorname{tors}}}(A[\ell],A[\ell]) \cong \nicefrac{\mathbb{Z}}{\ell \mathbb{Z}}$ for all but finitely many prime numbers $\ell$.

Let $P\in A[\ell^m]$ be of exact order $\ell^m$ and let $\varphi \in \operatorname{Hom}_{G_{\operatorname{tors}}}(\mathbb{Z}[G_{\operatorname{tors}}]\cdot P,A[\ell^m])$. We take any $P'\in A[\ell^m]$ such that $A[\ell^m] = \langle P , P' \rangle$. As in the proof of Lemma \[lem:fullorbit\] there is an element $\sigma \in G_{\operatorname{tors}}$ such that $\sigma(P)=P$ and $\sigma(P')=N\cdot P+ P'$ for some $0< N < \ell^{m_\ell}$.

Let $\varphi(P)=a\cdot P + b \cdot P'$ for some $a,b \in \mathbb{Z}$. Since $\varphi$ is a $G_{\operatorname{tors}}$-module homomorphism, it is $\varphi(\sigma(P))=\sigma(\varphi(P))$. This means that the following equality must hold $$a\cdot P + b \cdot P' = (a+bN)\cdot P + b \cdot P'.$$ Hence, $\ell^m \mid bN$ implying $\ell^{m-m_{\ell}+1} \mid b$. Therefore, there are at most $\ell^m \cdot \ell^{m_{\ell} -1}$ possible choices for $\varphi(P)$, and hence $\vert \operatorname{Hom}_{G_{\operatorname{tors}}}(\mathbb{Z}[G_{\operatorname{tors}}]\cdot P,A[\ell^m]) \vert \leq \ell^m \cdot \ell^{m_{\ell} -1}$. By Lemma \[lem:fullorbit\], there are at most $\ell^{2(m_{\ell} -1)}$ possibilities to extend an element from $\operatorname{Hom}_{G_{\operatorname{tors}}}(\mathbb{Z}[G_{\operatorname{tors}}]\cdot P,A[\ell^m])$ to an element in $\operatorname{Hom}_{G_{\operatorname{tors}}}(A[\ell^m],A[\ell^m])$. This proves the first statement of the Corollary.

Again the the second statement follows by noting that $m_{\ell}=1$ for almost all primes $\ell$.

Let $\Gamma \subseteq A(\overline{\mathbb{Q}})$ be a subgroup of rank $r$, with $r$ linearly independent elements $\gamma_1,\ldots,\gamma_r$. In sight of Theorem \[thm:freeab\] the goal is to prove that $\nicefrac{A(L)}{\Gamma_{\operatorname{div}}}$ is free abelian for any finite extension $L$ of $K(\Gamma_{\operatorname{div}})$. Note, that $\Gamma_{\operatorname{sat}} =\Gamma_{\operatorname{div}}$ in the present situation. Since the property of being free abelian is inherited by subgroups, we may enlarge the field $L$ as we please. In particular, we may assume $\gamma_1,\ldots,\gamma_r\in A(L)$.

After replacing our base field $K$ by a finite extension, we can also assume $L=K(\Gamma_{\operatorname{div}})$. Thanks to Lemma \[freemodgamma\] and Proposition \[prop:firstclaim\] it would be enough to prove that the torsion group of $\nicefrac{A(L)}{A(K')+\Gamma_{\operatorname{div}}}$ has finite exponent for all finite extensions $K'/K$. This is surely the case if the statement is true for $\nicefrac{A(K'L)}{A(K')+\Gamma_{\operatorname{div}}}$. Hence we would have to prove that the torsion group of $$\label{eq:group}
\nicefrac{A(K(\Gamma_{\operatorname{div}}))}{A(K)+\Gamma_{\operatorname{div}}}$$ has finite exponent for all sufficiently large number fields $K$ over which $A$ is defined. Hence, we fix an arbitrary number field $K$ such that $A$ is defined over $K$, and such that assumptions , and are satisfied.

\[prop:aaone\] For all but finitely many prime numbers $\ell$ the $\ell$-torsion part of the group from is trivial.

For all but finitely many prime numbers it is $A[\ell]$ a simple $G_{\operatorname{tors}}$-module with $$\operatorname{Hom}_{G_{\operatorname{tors}}}(A[\ell],A[\ell]) \cong \nicefrac{\mathbb{Z}}{\ell\mathbb{Z}}$$ (cf. Lemma \[lem:fullorbit\] and Corollary \[cor:homsmall\]). Let $\ell$ be such a prime number with the additional assumption that $\ell$ does not divide the exponent of the torsion group of $\nicefrac{A(K(A_{\operatorname{tors}}))}{A(K)+A_{\operatorname{tors}}}$. We denote this exponent (which is an integer by Theorem \[thm:BHP\]) by $c$.

For the sake of contradiction we assume that there is some $[\alpha]\in \nicefrac{A(K(\Gamma_{\operatorname{div}}))}{A(K)+\Gamma_{\operatorname{div}}}$ of order $\ell$. In particular, $\alpha \notin \Gamma_{\operatorname{div}}$. After replacing $\alpha$ by some element in $\alpha+ \Gamma_{\operatorname{div}}$, we may assume $\ell \cdot \alpha \in A(K)$. Since we assume $\ell \nmid c$, it is $\alpha \notin A(K(A_{\operatorname{tors}}))$. Hence $\gamma_1,\ldots,\gamma_r,\ell\cdot\alpha$ satisfy the properties , and , with $\Gamma$ replaced by $\langle\Gamma,\ell\cdot\alpha\rangle$. Hence, by Corollary \[cor:Galois\] it is $\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell}\langle \gamma_1,\ldots,\gamma_r,\ell\cdot\alpha \rangle)/K(A_{\operatorname{tors}}))$ isomorphic to a $G_{\operatorname{tors}}$-submodule of $A[\ell]^{r+1}$. Since $A[\ell]$ is simple by assumption, it follows $$\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell}\langle \gamma_1,\ldots,\gamma_r,\ell\cdot\alpha \rangle)/K(A_{\operatorname{tors}}))\cong A[\ell]^{r'}$$ with $r'\leq r+1$. Now Corollary \[cor:homsmall\] implies $$\vert \operatorname{Hom}_{G_{\operatorname{tors}}}(\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell}\langle \gamma_1,\ldots,\gamma_r,\ell\cdot\alpha\rangle)/K(A_{\operatorname{tors}})),A[\ell]) \vert = \ell^{r'}.$$ Since we also have $\ell^{r+1} \leq \vert \operatorname{Hom}_{G_{\operatorname{tors}}}(\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell}\langle \gamma_1,\ldots,\gamma_r,\ell\cdot\alpha\rangle)/K(A_{\operatorname{tors}})),A[\ell]) \vert$ by , it is $r'=r+1$. On the other hand, $\alpha$ is is $A(K(\Gamma_{\operatorname{div}}))$ and hence there is an $N\in\mathbb{N}$ such that $\alpha \in A(K(A_{\operatorname{tors}},\frac{1}{\ell^N} \langle\gamma_1,\ldots,\gamma_r\rangle))$ by Lemma \[lem:restriction\]. In particular, $\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell}\langle \gamma_1,\ldots,\gamma_r,\ell\cdot\alpha \rangle)/K(A_{\operatorname{tors}}))\cong A[\ell]^{r+1}$ is isomorphic to a quotient of $\operatorname{Gal}(K(A_{\operatorname{tors}},\frac{1}{\ell^N}\langle \gamma_1,\ldots,\gamma_r\rangle)/K(A_{\operatorname{tors}}))\subseteq A[\ell^N]^{r}$. This is a contradiction, and therefore there is no element of order $\ell$ in $\nicefrac{A(K(\Gamma_{\operatorname{div}}))}{A(K)+\Gamma_{\operatorname{div}}}$.

In order to give a full proof of Theorem \[thm:freeab\] for elliptic curves without complex multiplication, one needs the additional result that for all prime numbers $\ell$ the $\ell$-torsion part of the group from has finite exponent. This result remains open in this paper.

[*Acknowledgements:*]{} I would like to thank Gaël Rémond for providing the proof of Lemma \[Bogomolov\], which initiated this project, and for invaluable comments on an early version of this manuscript. Moreover, special thanks go to Arno Fehm for many interesting discussions on this topic, for pointing out the relevance of Bashmakov’s theorem, and for his patience reading first drafts of this manuscript.

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