The elliptic Lehmer problem for several independent algebraic points

Question

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?

Answer 1

ADDENDUM: I looked at Masser's 1989 article, and a quick back-of-the-envelope calculation seems to give the result, at least for two points. Thus if $P$ and $Q$ are independent points generating a field of degree $d$, then $$ \hat h(P)\hat h (Q) \ge \frac{C(E)}{d^3(\log d)^2}. $$ The idea is to apply the main theorem in Masser's paper to the set of points $$ \{ mP + nQ : 0\le m\lt M~\text{and}~0\le n\lt N \}. $$ Then choose the optimal values for $M$ and $N$, much as one does in proving the Cauchy-Schwarz inequality. Presumably one can do something similar with more points.

---

@Jamie: This is not Lang's height conjecture, which fixes a field (or degree of the field) and lets the elliptic curve vary. This is, as to OP indicates, a generalization of the elliptic version of Lehmer's conjecture.

@Vessilin: I remember corresponding with someone (maybe Cam Stewart) at one point about whether one could get a better bound for $\max \hat h(P_i)$, but we never got an interesting result. And I don't know anything about the problem of minimizing a product. Looks like an interesting problem. There are, as far as I know, three basic approaches that have been used for the $r=1$ case.

(1) For the CM case, Michel Laurent's estimate that mirrors Dobrowolski's.

(2) Use of Diophantine approximation / transcendence theory methods (i.e., auxiliary polynomials that become so small, they must vanish). The best estimate (up to improving the constant) is still Masser's and gives something like $\hat h(P) \ge C(E)/d^3(\log d)^2$.

(3) Use of Fourier averaging (which is the same idea that Hindry and I used for Lang's height conjecture). Hindry and I could only make this work if the $j$-invariant of $E$ is non-integral, but with this added hypothesis, we knocked one off of Masser's exponent and got $\hat h(P) \ge C(E)/d^2(\log d)^2$.

The reason I mention these is that it seems reasonable (to me) that each method has at least the possibility of being used to obtain a result for $\hat h(P_1)\hat h(P_2)\cdots\hat h(P_r)$.

There is also the following slightly different result: If you only look at points defined over the maximal abelian extension $K^{ab}$ of $K$, then there is an absolute upper bound $\hat h(P)\ge C(E)$. This was proven by combining two article, one by Matt Baker and one of mine, and then we have a joint paper that generalized it to abelian varieties. It is based on the ideas of Amoroso and Dvornicich, who proved the analogous result for the multiplicative group. The method can be roughly described as a careful use of ramification. Of course, in this case a product would have the obvious lower bound $C(E)^r$, so not very interesting. But I mention it since there are so few known methods that are potentially useful.