Unconditional generic solutions of equations involving the modular $j$-function and its iterates

For a system of polynomial equations involving the modular $j$-function, a solution is called generic when the transcendence degree of the coordinates it produces is as large as algebraically possible. Eterović and Herrero (Trans. Amer. Math. Soc. 374 (2021) 3971–3998) proved that such systems have Zariski-dense sets of solutions, and — assuming the modular Schanuel conjecture (MSC), a special case of the Grothendieck–André generalised period conjecture — that generic solutions exist for every irreducible plane curve that is not a horizontal or vertical line, and for equations involving the second iterate $j_2=j\circ j$. Unconditionally, generic solutions are known only for curves defined over $\overline{\mathbb{Q}}$ and, in later work, for varieties whose field of definition has no $C_j$-factors (a genericity condition on the base field, introduced by Eterović 2025, excluding fields containing the countable algebraically closed field $C_j$ built by solving systems of equations involving $j$); the existence (Zariski density) side is now largely unconditional, including for $j$ together with its first two derivatives, but density carries no transcendence-degree information. The open problem is to remove the dependence on MSC: prove unconditionally that every such curve over an arbitrary finitely generated field of definition contains infinitely many generic points $(z,j(z))$, extend genericity to all higher iterates $j_n$ (not available even conditionally for $n\ge 3$), and more fundamentally prove unconditional transcendence lower bounds relating $\mathrm{GL}_2^+(\mathbb{Q})$-orbit counts to transcendence degrees of tuples of $j$-values. A 2025 paper by Eterović states that even the (density-level) existential closedness problem for $j$ is still open, so the genericity core is firmly open as of 2026.

Background

The modular $j$-function is the holomorphic function on the complex upper half-plane $\mathbb{H}=\{z\in\mathbb{C}:\operatorname{Im}(z)>0\}$ with Fourier expansion $j(z)=q^{-1}+744+196884q+\cdots$ where $q=e^{2\pi i z}$; its value $j(z)$ is the $j$-invariant of the elliptic curve $\mathbb{C}/(\mathbb{Z}+z\mathbb{Z})$, so $j$ parametrizes isomorphism classes of complex elliptic curves. The group $G=\mathrm{GL}_2^{+}(\mathbb{Q})$ of rational $2\times 2$ matrices with positive determinant acts on $\mathbb{H}$ by fractional linear transformations $gz=\frac{az+b}{cz+d}$, and $j$ is invariant under the modular group $\mathrm{SL}_2(\mathbb{Z})\subset G$; two points of $\mathbb{H}$ in the same $G$-orbit carry algebraically related $j$-values, governed by the classical modular polynomials $\Phi_N$. A point $z\in\mathbb{H}$ is called special (or a CM point, for complex multiplication) when it is the unique fixed point in $\mathbb{H}$ of some element of $G$, equivalently when it satisfies a nontrivial quadratic equation over $\mathbb{Q}$; these are exactly the points whose associated elliptic curve has complex multiplication. A classical theorem of Schneider (1937) states that $\operatorname{tr.deg.}_{\mathbb{Q}}\mathbb{Q}(z,j(z))=0$ if and only if $z$ is special, where $\operatorname{tr.deg.}$ denotes the transcendence degree of a field extension; thus $j(z)$ is transcendental exactly at non-special (generic) points. Schanuel's conjecture for the exponential function predicts that $\operatorname{tr.deg.}_{\mathbb{Q}}\mathbb{Q}(z_1,\dots,z_n,e^{z_1},\dots,e^{z_n})\ge \operatorname{rank}_{\mathbb{Z}}(z_1,\dots,z_n)$; its modular analogue, the modular Schanuel conjecture (MSC), predicts that if $z_1,\dots,z_n\in\mathbb{H}$ are non-special points lying in pairwise distinct $G$-orbits, then $\operatorname{tr.deg.}_{\mathbb{Q}}\mathbb{Q}(z_1,\dots,z_n,j(z_1),\dots,j(z_n))\ge n$. Like classical Schanuel, MSC is a special case of the Grothendieck–André generalised period conjecture — the conjectural complete description of all algebraic relations among periods of algebraic varieties — and is wide open; the provable functional (differential) analogue is the Ax–Schanuel theorem for $j$ of Pila and Tsimerman (Duke Math. J. 165 (2016)), which yields the same shape of inequality for differentially independent inputs but gives no pointwise transcendence lower bounds for values. Eterović and Herrero (Trans. Amer. Math. Soc. 374 (2021) 3971–3998; arXiv:1907.09858) studied systems of equations $V\cap E_j^n\neq\emptyset$ where $V\subseteq\mathbb{C}^{2n}$ is an algebraic variety, $E_j^n=\{(z_1,\dots,z_n,j(z_1),\dots,j(z_n)):\mathbf{z}\in\mathbb{H}^n\}$ is the graph of $j$ on $\mathbb{H}^n$, and a point of $V$ of this form is generic over a field $K$ when $\operatorname{tr.deg.}_{K}K(\mathbf{z},j(\mathbf{z}))$ equals the dimension of the projection of $V$ (the maximal value not excluded by the equations). Their Theorem 1.1 gives, unconditionally, a Zariski-dense set of solutions whenever the projection of $V$ onto the $z$-coordinates is dominant; Theorem 1.2 shows, assuming MSC, that every irreducible plane curve $V\subset\mathbb{C}^2$ that is not a horizontal or vertical line has infinitely many points $(z,j(z))$ generic over any finitely generated field of definition $K$; for iterates $j_1=j$, $j_{n+1}=j\circ j_n$ (defined on the maximal domains $\mathbb{H}_n$ with $\mathbb{H}_{n+1}=\{z\in\mathbb{H}_n:j_n(z)\in\mathbb{H}\}$), Theorem 1.3 gives unconditional existence of solutions for all $n$, while Theorem 1.4 gives generic solutions under MSC only for the second iterate ($n=2$). The proof of Theorem 1.2 combines an unconditional counting result (Proposition 7.13: for $z$ with $j(z)$ algebraic of controlled degree and $g\in G$ with $N=\det g$, one has $[\mathbb{Q}(j(z),j(gz)):\mathbb{Q}(j(z))]\ge c_0N^{1/5}$, proved via modular polynomials and Pila-type counting — counting algebraic points on sets definable in an o-minimal structure, i.e. a model-theoretically 'tame' class of real sets with strong finiteness properties) with Lemma 7.15, a finiteness statement — non-special points of the curve lying in $\overline{K}\cap\mathbb{H}$ fall into finitely many $G$-orbits — which is the single conditional input and follows from MSC. For the exponential function the analogous genericity results rest on Baker's theorem on linear forms in logarithms; no comparable tool is known for $j$. Unconditionally, the source paper proves generic points only for curves defined over $\overline{\mathbb{Q}}$ (Proposition 7.6, via Schneider's theorem). The authors explicitly acknowledge as open: removing the MSC dependence (equivalently, proving unconditional transcendence lower bounds relating $G$-orbit counts to transcendence degrees of $j$-values), obtaining stronger unconditional results with more sophisticated tools such as o-minimality, extending genericity to higher iterates, and sharpening the arithmetic bounds.

Problem Statement

Prove, without any unproved hypothesis, the generic-solution statements that Eterović–Herrero (Trans. Amer. Math. Soc. 374 (2021) 3971–3998) currently obtain only under the modular Schanuel conjecture, and extend them to all higher iterates. Specifically: (1) For every irreducible algebraic curve $V\subset\mathbb{C}^2$ that is not a horizontal or vertical line and every finitely generated field $K\subset\mathbb{C}$ over which $V$ is defined, show that $V$ contains infinitely many points of the form $(z,j(z))$ with $z\in\mathbb{H}$ and $\operatorname{tr.deg.}_{K}K(z,j(z))=1$, with no assumption beyond ZFC — removing the MSC hypothesis from their Theorem 1.2, which is currently unconditional only for $V$ defined over $\overline{\mathbb{Q}}$ or over fields with no $C_j$-factors. (2) For iterates $j_n$ ($j_1=j$, $j_{n+1}=j\circ j_n$, with maximal domains $\mathbb{H}_n$): establish genericity results at the level of their Theorem 1.4 for every $n\ge 2$ — the unconditional statement for every $n$ (not currently available for $n\ge 3$ even conditionally on MSC): for hypersurfaces $V\subset\mathbb{C}^{n+1}$ defined by an irreducible polynomial $p(X,Y_1,\dots,Y_n)\in\overline{\mathbb{Q}}[X,Y_1,\dots,Y_n]$ with $\partial p/\partial X,\partial p/\partial Y_n\neq 0$, infinitely many $z\in\mathbb{H}_n$ satisfy $(z,j(z),\dots,j_n(z))\in V$ with $\operatorname{tr.deg.}_{\mathbb{Q}}\mathbb{Q}(z,j(z),\dots,j_n(z))=n$. (3) As the underlying transcendence input, prove unconditional lower bounds relating $G$-orbit counts to transcendence degrees of values of $j$ over finitely generated fields; that is, the full modular Schanuel lower bound — for non-special $z_1,\dots,z_n$ in pairwise distinct $G$-orbits, $\operatorname{tr.deg.}_{\mathbb{Q}}\mathbb{Q}(z_1,\dots,z_n,j(z_1),\dots,j(z_n))\ge n$ — proved for all $n$ with no hypothesis beyond ZFC. The finiteness statement of their Lemma 7.15 (non-special points of a curve in $\overline{K}\cap\mathbb{H}$ fall into finitely many $G=\mathrm{GL}_2^{+}(\mathbb{Q})$-orbits) and weak cases of MSC are partial inputs that fall short of this bound. Items (1)–(3) are one scientific objective and are jointly required: unconditional transcendence-degree genericity of solutions of equations involving $j$ and its iterates. The systematic characterization of which other automorphic or transcendental functions admit analogous theorems, and the sharpening of the effective determinant/level bounds, are not part of this objective.

References

1. Sebastian Eterović and Sebastián Herrero, 'Solutions of equations involving the modular $j$ function', Transactions of the American Mathematical Society 374 (2021) 3971–3998. DOI: 10.1090/tran/8244; arXiv:1907.09858, https://arxiv.org/abs/1907.09858
2. Sebastian Eterović, 'Generic solutions of equations involving the modular $j$ function', Mathematische Annalen 391 (2025) 6401–6449. DOI: 10.1007/s00208-024-03082-6; arXiv:2209.12192, https://arxiv.org/abs/2209.12192
3. Vahagn Aslanyan, Sebastian Eterović, and Vincenzo Mantova, 'Equations involving the modular $j$-function and its derivatives', Journal für die reine und angewandte Mathematik (Crelle's Journal), published online 2025. DOI: 10.1515/crelle-2025-0067; arXiv:2312.09974, https://arxiv.org/abs/2312.09974
4. Vahagn Aslanyan, Sebastian Eterović, and Jonathan Kirby, 'A closure operator respecting the modular $j$-function', Israel Journal of Mathematics 253 (2022) 321–357. DOI: 10.1007/s11856-022-2362-y; arXiv:2010.00102, https://arxiv.org/abs/2010.00102
5. Jonathan Pila and Jacob Tsimerman, 'Ax–Schanuel for the $j$-function', Duke Mathematical Journal 165 (2016), no. 13, 2587–2605. DOI: 10.1215/00127094-3620005; arXiv:1412.8255, https://arxiv.org/abs/1412.8255
6. Sebastian Eterović and Roy Zhao, 'Algebraic varieties and automorphic functions', International Mathematics Research Notices 2025 (2025), no. 4. DOI: 10.1093/imrn/rnaf029; arXiv:2107.10392, https://arxiv.org/abs/2107.10392
7. Sebastian Eterović and Adele Padgett, 'Some equations involving the gamma function', Transactions of the American Mathematical Society 378 (2025) 3445–3469. DOI: 10.1090/tran/9361; arXiv:2310.01658, https://arxiv.org/abs/2310.01658
8. Isaac A. Broudy and Sebastian Eterović, 'Schanuel Type Conjectures and Disjointness', preprint (2022). arXiv:2211.09556, https://arxiv.org/abs/2211.09556
9. Theodor Schneider, 'Arithmetische Untersuchungen elliptischer Integrale', Mathematische Annalen 113 (1937) 1–13. DOI: 10.1007/BF01571618, https://doi.org/10.1007/BF01571618