{"paper":{"title":"$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $\\Gamma_1(4)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jigu Kim, Yoonjin Lee","submitted_at":"2021-09-17T22:47:34Z","abstract_excerpt":"For a prime $p\\equiv 3$ $(\\text{mod }4)$ and $m\\ge 2$, Romik raised a question about whether the Taylor coefficients around $\\sqrt{-1}$ of the classical Jacobi theta function $\\theta_3$ eventually vanish modulo $p^m$. This question can be extended to a class of modular forms of half-integral weight on $\\Gamma_1(4)$ and CM points; in this paper, we prove an affirmative answer to it for primes $p\\ge5$. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on $\\mathrm{SL}_2(\\mathbb{Z})$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.08778","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2109.08778/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}