MathDB

Existential closedness conjecture for the modular jj-function

About 6 years old · traced to

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 1Other wordings

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).

References

Primary source

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

existential-closednessmodel-theorymodular-curvesmodular-functionsunlikely-intersections
Posted

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.