Consider the following conjecture, going back to Lang and restated (and proved) in the elliptic case in a 2009 Crelle paper by David and Hirata-Kohno (see Conjecture 1.2 in their paper).

Conjecture (Lang): Let $A$ be an abelian variety defined over a number field $K$ and $\varphi$ a non-constant rational function on $A$ defined over $K$. Let $h$ be a logarithmic height on $A$ with respect to some projective embedding of $A$. Then there exist positive constants $C, N$ such that $$|\varphi(P)| \ge h(P)^{−C}$$ for every $P \in A(K)$ satisfying $\varphi(P) \ne 0$ and $h(P) \ge N$.

Lang explains (for instance in his book Number theory III) that if $A$ is one-dimensional, then this conjecture can be reduced to a statement about linear forms in elliptic logarithms. Indeed, in this case the conjecture can be proved by deducing it from a lower bound on linear forms in elliptic logarithms proved in the above-mentioned paper of David-Hirata-Kohno.

I'm interested in the higher-dimensional case.

Question 1: If $A$ has dimension $>1$, can the conjecture also be reduced to a statement about linear forms in abelian logarithms?

Gaudron has proved lower bounds on linear forms in abelian logarithms which (for our purposes) generalize the bounds of David-Hirata-Kohno.

Question 2: Does the conjecture follow from Gaudron's results? If not, are any partial results known in the higher-dimensional case?



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