{
  "abstract": "We prove the Modular Existential Closedness conjecture for the modular $j$-function extended from the upper half-plane to $\\mathbb{C} \\setminus \\mathbb{R}$ by Schwarz reflection. Specifically, we show that any system of equations involving the field operations and $j$ has solutions in the complex numbers unless it is overdetermined. Such a system is overdetermined if, roughly, there are more independent equations than variables, possibly after applying the transformation rules of the $j$-function under the linear fractional action of $\\rm{PGL}_2(\\mathbb{Q})$ on $\\mathbb{C} \\setminus \\mathbb{R}$. In general, we do not expect overdetermined systems of equations to have solutions due to the algebraic and transcendental properties of the $j$-function captured by the modular analogue of Schanuel's conjecture. In geometric terms, Modular Existential Closedness states that given a broad and free irreducible algebraic variety $V \\subseteq \\mathbb{C}^n \\times \\mathbb{C}^n$ -- where broadness and freeness are geometric conditions guaranteeing that the system is not overdetermined -- there is a point of the form $(τ_1, \\ldots, τ_n, j(τ_1), \\ldots, j(τ_n)) \\in V$ with $τ_i \\in \\mathbb{C} \\setminus \\mathbb{R}$. The proof uses Ratner's theorems on the closure and equidistribution of the orbits of unipotent flows in powers of $\\rm{PGL}_2(\\mathbb{R})$ to describe the asymptotic behaviour of the orbits of certain analytic maps. This is then applied to approximate a relatively simple system of equations using the original system and the action of $\\rm{PGL}_2(\\mathbb{Z})$. The construction of the limiting system makes use of the Ax--Schanuel theorem and guarantees an isolated solution. Rouché's theorem then establishes the existence of solutions for the original system of equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35409",
  "author_total": 1,
  "authors": [
    {
      "name": "Vahagn Aslanyan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4btavbs5hr6c4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35409",
  "fulltext_url": null,
  "id": "4btavbs5hr6c4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35409",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223947138",
  "pdf_url": "https://arxiv.org/pdf/2610.35409",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4btavbs5hr6c4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.35409",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35409"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Modular Existential Closedness",
  "title_text": "Modular Existential Closedness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/4btavbs5hr6c4",
  "venue": "arXiv",
  "versions": []
}