{"paper":{"title":"Eisenstein series of weight one, $q$-averages of the $0$-logarithm and periods of elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Daniel R. Grayson, Dinakar Ramakrishnan","submitted_at":"2018-06-13T10:12:18Z","abstract_excerpt":"For any elliptic curve $E$ over $k\\subset \\Bbb R$ with $E({\\Bbb C})={\\Bbb C}^\\times/q^{\\Bbb Z}$, $q=e^{2\\pi iz}, \\Im(z)>0$, we study the $q$-average $D_{0,q}$, defined on $E({\\Bbb C})$, of the function $D_0(z) = \\Im(z/(1-z))$. Let $\\Omega^+(E)$ denote the real period of $E$. We show that there is a rational function $R \\in {\\Bbb Q}(X_1(N))$ such that for any non-cuspidal real point $s\\in X_1(N)$ (which defines an elliptic curve $E(s)$ over $\\Bbb R$ together with a point $P(s)$ of order $N$), $\\pi D_{0,q}(P(s))$ equals $\\Omega^+(E(s))R(s)$. In particular, if $s$ is $\\Bbb Q$-rational point of $X"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.04925","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"}