{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-landau-siegel-zeros-and-heights-of","title":"On Landau-Siegel zeros and heights of singular moduli","arxiv_id":"1911.07215","date":"2019-11-17","proceeding":null,"authors":["Christian Táfula"],"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 $D$.","url_abs":"https://arxiv.org/abs/1911.07215v3","url_pdf":"https://arxiv.org/pdf/1911.07215v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-landau-siegel-zeros-and-heights-of","repo_url":"https://github.com/tafula/loght_jtd","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}