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The higher dimensional Lehmer problem asserts that if $\alpha_1,\ldots,\alpha_r$ are multiplicatively independent non-zero algebraic numbers generating an extension of $\mathbb{Q}$ of degree $d$, then $h(\alpha_1) \cdots h(\alpha_r) \geq c(r)/d$. This arose in the work of Amoroso and David, who have shown this up to a logarithmic factor, in a generalization of Dobrowolski's bound. As a corollary, Amoroso and David obtain the truth of the original Lehmer conjecture under the assumption that $\alpha$ generates a normal extension of $\mathbb{Q}$.

The natural analog for the case of elliptic curves ought to be:

If $P_1,\ldots,P_r \in E(\bar{\mathbb{Q}})$ are independent algebraic points generating a number field of degree $d$, then the product of their canonical heights is at least $c(E,r)/d$.

Still, I have not yet seen this statement explicitly mentioned in the literature. (Is it indeed expected to hold?)

Usually, results on the original Lehmer problem transfer without great difficulties over to the case of CM elliptic curves (essentially because the CM hypothesis allows to lift Frobenii). For instance, the literal analog of Dobrowolski's bound for CM elliptic curves is due to Laurent, from 1981.

Question. In light of this, has the above statement about $r$ independent algebraic points on a CM elliptic curve been proved up to a logarithmic factor? In particular, is the Lehmer conjecture known to be true for points on CM elliptic curves that generate a normal extension of $\mathbb{Q}$? Are there any results available in the literature?

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# The elliptic Lehmer problem for several independent algebraic points

The higher dimensional Lehmer problem asserts that if $\alpha_1,\ldots,\alpha_r$ are multiplicatively independent non-zero algebraic numbers generating an extension of $\mathbb{Q}$ of degree $d$, then $h(\alpha_1) \cdots h(\alpha_r) \geq c(r)/d$. This arose in the work of Amoroso and David, who have shown this up to a logarithmic factor, in a generalization of Dobrowolski's bound. As a corollary, Amoroso and David obtain the truth of the original Lehmer conjecture under the assumption that $\alpha$ generates a normal extension of $\mathbb{Q}$.

The natural analog for the case of elliptic curves ought to be:

If $P_1,\ldots,P_r \in E(\bar{\mathbb{Q}})$ are independent algebraic points generating a number field of degree $d$, then the product of their canonical heights is at least $c(E,r)/d$.

Still, I have not yet seen this statement explicitly mentioned in the literature. (Is it indeed expected to hold?)

Usually, results on the original Lehmer problem transfer without great difficulties over to the case of CM elliptic curves (essentially because the CM hypothesis allows to lift Frobenii). For instance, the literal analog of Dobrowolski's bound for CM elliptic curves is due to Laurent, from 1981.

Question. In light of this, has the above statement about $r$ independent algebraic points on a CM elliptic curve been proved? Are there any results available in the literature?