{"paper":{"title":"Algebraic integers as special values of modular units","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dong Hwa Shin, Dong Sung Yoon, Ja Kyung Koo","submitted_at":"2010-08-03T07:29:52Z","abstract_excerpt":"Let $\\varphi(\\tau)=\\eta((\\tau+1)/2)^2/\\sqrt{2\\pi}e^\\frac{\\pi i}{4}\\eta(\\tau+1)$ where $\\eta(\\tau)$ is the Dedekind eta-function. We show that if $\\tau_0$ is an imaginary quadratic number with $\\mathrm{Im}(\\tau_0)>0$ and $m$ is an odd integer, then $\\sqrt{m}\\varphi(m\\tau_0)/\\varphi(\\tau_0)$ is an algebraic integer dividing $\\sqrt{m}$. This is a generalization of Theorem 4.4 given in [B. C. Berndt, H. H. Chan and L. C. Zhang, Ramanujan's remarkable product of theta-functions, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 3, 583-612]. On the other hand, let $K$ be an imaginary quadratic field and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.0473","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}