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Existential closedness conjecture for the modular jj-function

Let H⁡\operatorname{\mathbb{H}} be the complex upper half-plane, let j:H⁡→C⁡j:\operatorname{\mathbb{H}}\to\operatorname{\mathbb{C}} be the modular jj-function, and let Γj={(zˉ,j(zˉ)):zˉ∈H⁡n}⊆C⁡2n\Gamma_j=\{(\bar z,j(\bar z)):\bar z\in\operatorname{\mathbb{H}}^n\}\subseteq\operatorname{\mathbb{C}}^{2n}. A variety is called jj-broad, jj-free, or H⁡\operatorname{\mathbb{H}}-free according to the properties defined in the source. Existential closedness conjecture for jj. If V⊆H⁡n×C⁡nV\subseteq\operatorname{\mathbb{H}}^n\times\operatorname{\mathbb{C}}^n is an irreducible jj-broad, jj-free and H⁡\operatorname{\mathbb{H}}-free variety defined over C⁡\operatorname{\mathbb{C}}, then

V∩Γj≠∅.V\cap\Gamma_j\neq\emptyset.

This is the dual existence statement to the Modular Schanuel conjecture: every system that is not overdetermined for transcendence-theoretic reasons should have a complex solution. The supplied source proves related existential-closedness results for blurred versions of jj, but gives no resolution of this conjecture itself.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The existential closedness conjecture for the modular j-function

    Let n≥1n\geq 1, let Y(1)\mathrm{Y}(1) denote the modular curve, and let

    Ejn={(z1,…,zn,j(z1),…,j(zn)):z1,…,zn∈H}.\mathrm{E}_j^n=\{(z_1,\ldots,z_n,j(z_1),\ldots,j(z_n)):z_1,\ldots,z_n\in\mathbb{H}\}.

    A variety in Cn×Y(1)n\mathbb{C}^n\times\mathrm{Y}(1)^n is broad when every coordinate projection indexed by ℓ\ell coordinates has dimension at least ℓ\ell, and it is free in the sense used in the paper. Existential closedness conjecture. If V⊆Cn×Y(1)nV\subseteq\mathbb{C}^n\times\mathrm{Y}(1)^n is broad and free, then

    V∩Ejn≠∅.V\cap\mathrm{E}_j^n\neq\emptyset.

    Known results establish this for important classes, including varieties whose first projection is Zariski dense, but the general broad-and-free case is presented as conjectural.

    source: Sebastian Eterović, “Generic Solutions of Equations Involving the Modular j-function”, arXiv:2209.12192 (2025).

Sources & referencesView supporting material

Primary source

Vahagn Aslanyan and Jonathan Kirby, “Blurrings of the j-function”, arXiv:2005.10167 (2021).

existential-closednessmodel-theorymodular-curvesmodular-functionsunlikely-intersections

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