[
    {
        "id": "authors:kpwma-p9093",
        "collection": "authors",
        "collection_id": "kpwma-p9093",
        "cite_using_url": "https://authors.library.caltech.edu/records/kpwma-p9093",
        "type": "article",
        "title": "The unbounded denominators conjecture",
        "author": [
            {
                "family_name": "Calegari",
                "given_name": "Frank",
                "orcid": "0000-0001-5542-6373"
            },
            {
                "family_name": "Dimitrov",
                "given_name": "Vesselin",
                "orcid": "0000-0002-1515-8981",
                "clpid": "Dimitrov-Vesselin"
            },
            {
                "family_name": "Tang",
                "given_name": "Yunqing",
                "orcid": "0000-0002-5811-4196"
            }
        ],
        "abstract": "<p>We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of SL<span class=\"diff-html-added\">\u2082</span>(Z). Our result includes also Mason's generalization of the original conjecture to the setting of vector-valued modular forms, thereby supplying a new path to the congruence property in rational conformal field theory. The proof involves a new arithmetic holonomicity bound of a potential-theoretic flavor, together with Nevanlinna second main theorem, the congruence subgroup property of SL\u2082(Z[1/p]), and a close description of the Fuchsian uniformization D(0,1)/&Gamma;N of the Riemann<span class=\"diff-html-added\">\u2082</span> surface C\u2216&mu;N.</p>",
        "doi": "10.1090/jams/1053",
        "issn": "1088-6834",
        "publisher": "American Mathematical Society (AMS)",
        "publication": "Journal of the American Mathematical Society",
        "publication_date": "2025-07",
        "series_number": "3",
        "volume": "38",
        "issue": "3",
        "pages": "627-702"
    }
]