An effective matrix lemma
Masser's matrix lemma gives a lower bound of the height of an abelian variety in terms of its period lattices. Precisely, let \((A;L)\) be a principally polarized abelian variety of dimension \(g\) defined over a number field \(K\) and let \(\kappa_1\) be the constant \(\frac{1}{2}\ln\frac{2\pi^3e}{3}\). Set \(\rho_\sigma=\min(\rho(A_\sigma;L_\sigma),\sqrt{\frac{\pi}{3g}})\) where \(\sigma\in G_K:=\Hom (K,\mathbb{C})\) and \(\rho(A_\sigma;L_\sigma)\) is the injectivity diameter (i.e., the first minimum of period lattice) associated to a principally polarized complex abelian variety. Then one has the inequality \[ h_{\text{Fa}}(A)+\kappa_1g\geq\frac{1}{[K:\mathbb{Q}]}\sum_{\sigma\in G_K}(\frac{\pi}{6\rho_\sigma^2}+g\ln(\rho_\sigma\sqrt{g})). \] The purpose of the paper under review is to give an almost optimal effective proof of Masser's matrix lemma. The main result can be described as follows. To a principally polarized complex abelian variety \((A;L)\), the author fixes a Haar measure \(v_1\) of volume \(1\) on \(A(\mathbb{C})\) and chooses an invertible sheaf \(\mathcal{L}\) defining \(L\), a cubic metric \(\|\cdot\|\) on \(\mathcal{L}\) and a nonzero section \(s\) of \(\mathcal{L}\). Then the integral \[ I(A;L)=-\int_{A(\mathcal{C})}\ln\|s\|{\text{d}}v_1+\frac{1}{2}\ln\int_{A(\mathcal{C})}\|s\|^2{\text{d}}v_1 \] is real and independent of the choice of \((\mathcal{L};\|\cdot\|;s)\). Let \(\rho=\min(\rho(A;L),\sqrt{\frac{\pi}{3g}})\), the author showed that \[ I(A;L)\geq\frac{\pi}{12\rho^2}+\frac{g}{2}\ln\rho+\frac{g}{4}\ln\frac{3g}{\pi e}. \] This result combined with the following inequality \[ h_{\text{Fa}}(A)\geq-\frac{g}{2}\ln(2\pi^2)+\frac{2}{[K:\mathbb{Q}]}\sum_{\sigma\in G_K}I(A_\sigma;L_\sigma), \] which is due to J.-B. Bost, implies Masser's matrix lemma.
scientific article; zbMATH DE number 6134710
scientific article