On Landau-Siegel zeros and heights of singular moduli

Christian Táfula

Published 2019-11-17Version 1

Let $D < 0$ be a fundamental discriminant, $\chi_D$ the Dirichlet character associated to $\mathbb{Q}(\sqrt{D})$, and $\tau_D := i\sqrt{|D|}/2$ if $D \equiv 0 \,(\mathrm{mod}~4)$ or $\tau_D := (-1+i\sqrt{|D|})/2$ if $D \equiv 1 \,(\mathrm{mod}~4)$. Based on the work of Granville-Stark [DOI:10.1007/s002229900036], and a theorem of Duke [DOI:10.1007/BF01393993] on the uniform distribution of Heegner points, we show that \[ \mathrm{ht}(j(\tau_D)) = 6\, \Bigg(\sum_{\substack{0 < \mathrm{Re}(\varrho) < 1 \\ L(\varrho,\chi_D)=0}} \frac{1}{\varrho} \Bigg) + C + o(1) \] as $D\to -\infty$, where $j$ is the $j$-invariant function, $\mathrm{ht}$ is the absolute logarithmic Weil height, and $C \approx 11.511550\ldots$. From that, we measure the effect of the largest real zero of $L(s,\chi_D)$ (say, $\beta_D$) to the growth of $\mathrm{ht}(j(\tau_D))$, allowing us to obtain, from the uniform $abc$-conjecture for number fields, the estimate \[ \beta_D \leq 1 - \frac{10/(5-\sqrt{5}) + o(1)}{\log(|D|)}, \] where $10/(5-\sqrt{5}) \approx 3.618033\ldots$.