Lehmer-type bounds and counting rational points of bounded heights on abelian varieties (Q6591610)

Let \(K\) be a number field. Let \(A/K\) be an abelian variety of dimension \(g\). Let \(\hat h \) be the Néron-Tate height on \(A(\bar K)\) associated with a symmetric ample line bundle \(\mathcal L\) on \(A\). The following is known as Lehmer Conjecture, and there is rich literature on the bounds related to the conjecture: there is a positive constant \(C=C(A,K)\) such that \(\hat h(P)\ge C / D^{1/g}\) for all non-torsion point \(P\in A(\bar K)\) where \(D\) is the field extension degree \([K(P):K]\). The authors of the paper under review prove that \(\hat h(P) \ge C/(D \log D)^{2g}\) if \(D\ge 3\) and \(K(P)\) is Galois over \(K\), and that given \(\varepsilon>0\), \(\hat h(P) \ge C/D^{2g+\varepsilon}\) if \(D\ge 1\) and \(K(P)\) is Galois over \(K\).\N\NThe authors also introduce results on the number of rational points on \(A/K\). Let \(H_K\) be the multiplicative height on \(A(K)\), and let \(\hat h_K\) be the Néron-Tate height on \(A( K)\) associated with a symmetric ample line bundle \(\mathcal L\) on \(A/K\). They prove that there are positive constants \(C\) and \(C'\) such that \N\[\N\#\{ P \in A(K) : \hat h_K(P) \le \log B\} \le B^{C/\log\log B} , \quad \#\{ P \in A(K) : H_K(P) \le B\} \le B^{C'/\log\log B} \]\Nfor all \(B\ge e^e\), provided that \(A\) has a \(K\)-rational \(p\)-torsion point where \(p\) is a rational prime. There are quite a few results in the literature on counting the number of rational points if \(g=1\). The authors of the paper under review show that the constants \(C\) and \(C'\) of their results are computable if \(A\) is the product of elliptic curves over \(K\).