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 \[ \frac{L'}{L}(1,χ_D) = \frac{1}{6}\, \mathrm{height}(j(τ_D)) - \frac{1}{2}\log|D| + C + o_{D\to -\infty}(1), \] where ${\tau }_D=\frac{1}{2}(-\delta +\sqrt{D})$ for $D\equiv \delta (\mathrm{mod}4)$ and $j(\cdot )$ is the $j$-invariant function with $C=-1.057770\dots$. Assuming the ``uniform'' $abc$-conjecture for number fields, we deduce that $L(\beta ,{\chi }_D)\ne 0$ with $\beta \ge 1-\frac{\sqrt{5}\phi +o(1)}{\log |D|}$ where $\phi =\frac{1+\sqrt{5}}{2}$, which we improve for smooth $D$.

25 pages, 2 figures. Previous uploads [v1], [v2] = thesis version. Current upload [v3] = expanded journal version. For the scripts used to generate the figures, see https://github.com/tafula/loght_jtd

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