Rémond's generalized Lehmer conjecture

Let $\mathcal{G}$ be either a torus or an abelian variety defined over a number field $K$, and let $\Gamma$ be a finite rank subgroup of $\mathcal{G}(\overline{\mathbb{Q}})$. Write $\Gamma_{\operatorname{sat}}$ for the saturation of $\Gamma$ under the endomorphisms of $\mathcal{G}$ and divisibility, and call $\alpha\in\mathcal{G}(\overline{\mathbb{Q}})$ $\Gamma$-transversal if it lies in a translate $\gamma+B$ with $\gamma\in\Gamma_{\operatorname{sat}}$ and $B$ a connected proper algebraic subgroup of $\mathcal{G}$. Let $\widehat{h}_{\mathcal{G}}$ denote the canonical height and let $\dim(\mathcal{G})$ be the dimension of $\mathcal{G}$.

Rémond's generalized Lehmer conjecture. The following assertions hold:

(a) There exists a positive constant $c$ such that

$$\widehat{h}_{\mathcal{G}}(\alpha)\geq \frac{c}{[K(\Gamma_{\operatorname{sat}})(\alpha):K(\Gamma_{\operatorname{sat}})]^{1/\dim(\mathcal{G})}}$$

for every $\alpha\in\mathcal{G}(\overline{\mathbb{Q}})$ that is not $\Gamma$-transversal.

(b) For every $\varepsilon>0$ there exists a positive constant $c_{\varepsilon}$ such that

$$\widehat{h}_{\mathcal{G}}(\alpha)\geq \frac{c_{\varepsilon}}{[K(\Gamma_{\operatorname{sat}})(\alpha):K(\Gamma_{\operatorname{sat}})]^{1/\dim(\mathcal{G})+\varepsilon}}$$

for every $\alpha\in\mathcal{G}(\overline{\mathbb{Q}})$ that is not $\Gamma$-transversal.

(c) For every finite extension $L/K(\Gamma_{\operatorname{sat}})$ there exists a positive constant $c_L$ such that

$$\widehat{h}_{\mathcal{G}}(\alpha)\geq c_L$$

for every $\alpha\in\mathcal{G}(L)\setminus\Gamma_{\operatorname{sat}}$.