{"paper":{"title":"Congruences involving the $U_{\\ell}$ operator for weakly holomorphic modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dohoon Choi, Subong Lim","submitted_at":"2019-02-18T08:31:49Z","abstract_excerpt":"Let $\\lambda$ be an integer, and $f(z)=\\sum_{n\\gg-\\infty} a(n)q^n$ be a weakly holomorphic modular form of weight $\\lambda+\\frac 12$ on $\\Gamma_0(4)$ with integral coefficients. Let $\\ell\\geq 5$ be a prime. Assume that the constant term $a(0)$ is not zero modulo $\\ell$. Further, assume that, for some positive integer $m$, the Fourier expansion of $(f|U_{\\ell^m})(z) = \\sum_{n=0}^\\infty b(n)q^n$ has the form \\[ (f|U_{\\ell^m})(z) \\equiv b(0) + \\sum_{i=1}^{t}\\sum_{n=1}^{\\infty} b(d_i n^2) q^{d_i n^2} \\pmod{\\ell}, \\] where $d_1, \\ldots, d_t$ are square-free positive integers, and the operator $U_\\e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.06456","kind":"arxiv","version":1},"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"}