{"paper":{"title":"$q$-deformation with ($\\varphi, \\Gamma$) structure of the de Rham cohomology of the Legendre family of elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Ryotaro Shirai","submitted_at":"2020-06-22T14:45:12Z","abstract_excerpt":"In the late '60s, B. Dwork studied a Frobenius structure compatible with the classical hypergeometric differential equation with parameters $\\left(\\frac{1}{2},\\frac{1}{2} ; 1 \\right)$ by analyzing behavior of solutions of the differential equation under Frobenius transformation. Recently, P. Scholze conjectured the existence of $q$-de Rham cohomology groups for any $\\mathbb{Z}$-scheme. In this paper, we give a Frobenius structure compatible with the $q$-hypergeometric differential equation with parameters $(q^{\\frac12},q^{\\frac12};q)$ by showing a $q$-analogue of some results of Dwork. This co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.12310","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.12310/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"}