On Landau-Siegel zeros and heights of singular moduli

Abstract

Let $\chi_D$ be the Dirichlet character associated to $\mathbb{Q}(\sqrt{D})$ where $D < 0$ is a fundamental discriminant. Improving Granville-Stark [DOI:10.1007/s002229900036], we show that \[ \frac{L'}{L}(1,\chi_D) = \frac{1}{6}\, \mathrm{height}(j(\tau_D)) - \frac{1}{2}\log|D| + C + o_{D\to -\infty}(1), \] where $\tau_D = \frac 12(-\delta+\sqrt{D})$ for $D \equiv \delta ~(\mathrm{mod}~4)$ and $j(\cdot)$ is the $j$-invariant function with $C = -1.057770\ldots$. Assuming the ``uniform'' $abc$-conjecture for number fields, we deduce that $L(\beta,\chi_D)\ne 0$ with $\beta \geq 1 - \frac{\sqrt{5}\varphi + o(1)}{\log|D|}$ where $\varphi = \frac{1+\sqrt{5}}{2}$, which we improve for smooth $D$.