Author’s summary: “We obtain a lower bound for the normalised height of a non-torsion hypersurface \(V\) of a C.M.abelian variety \(A\) which is a refinement of a precedent result. This lower bound is optimal in terms of the geometric degree of \(V\), up to an absolute power of a “log” (independent of the dimension of \(A\)). We thus extend the results of F. Amoroso and S. David [Acta Arith. 92, No. 4, 339–366 (2000; Zbl 0948.11025)] on the same problem on a multiplicative group \({\mathbb G}_m^n\). When \(A\) is an elliptic curve and \(V=\{\overline{P}\}\) is the set of conjugates of a non-torsion \(\overline{k}\)-point, we reobtain the result of M. Laurent on the elliptic Lehmer’s problem.”
His main result is stated as Theorem 3: Let \(A\) be a CM abelian variety, \({\mathcal L}\) an ample and symmetric line bundle of \(A\) and \(V\) is an irreducible hypersurface of \(A\) over \(k\) that is not the union of torsion subvarieties, then the following inequality holds: \[ \widehat {h}_{\mathcal L}(V)\geq \deg_{\mathcal L}(V)\widehat\mu^{\text{ess}}_{\mathcal L}(V)\geq c(A/k,{\mathcal L})\frac{(\log \log \deg_{\mathcal L}(V))^{{1+2\delta}_ {g-s,1}}}{(\log \deg _{\mathcal L}(V))^{{2+\delta}_{g-s,1}}}, \] where \(s\) is the dimension of the stabilizer of \(V\).