The Zilber–Pink conjecture for powers of the modular curve $Y(1)^n$

The Zilber–Pink conjecture predicts that every subvariety $W$ of $Y(1)^n$ — the $n$-fold self-product of the affine modular curve coordinatized by the modular $j$-function — contains only finitely many maximal atypical subvarieties, i.e. maximal components of intersections with special subvarieties that are larger than dimension counting predicts. Pila and Tsimerman's Ax–Schanuel theorem for the $j$-function proved this for strongly atypical subvarieties, those on which no coordinate is constant; the cases with constant (singular-modulus) coordinates have remained open, blocked by the large Galois orbits problem for optimal points. The two companion Ax–Schanuel conjectures posed in the same paper — for the uniformization of the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties, and with derivatives for arbitrary Shimura varieties — were subsequently resolved by Mok, Pila, and Tsimerman, and the geometrically generic (non-$\overline{\mathbb{Q}}$-defined) case of Zilber–Pink for $Y(1)^n$ was settled in 2025. The surviving open core is the arithmetic case: prove the conjecture for subvarieties with coordinate projections defined over $\overline{\mathbb{Q}}$.

Background

The modular $j$-function $j(z)$ is the basic holomorphic invariant of the upper half-plane $\mathbb{H}$ under the action of $\mathrm{SL}_2(\mathbb{Z})$ by fractional linear transformations: it is $\mathrm{SL}_2(\mathbb{Z})$-invariant and identifies the quotient $Y(1)=\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}$, the coarse moduli space of complex elliptic curves, with the affine line via the $j$-invariant. The points $j(\tau)$ with $\tau$ imaginary quadratic are called singular moduli; they are the $j$-invariants of elliptic curves with complex multiplication (endomorphisms beyond $\mathbb{Z}$) and are algebraic numbers. For a product $Y(1)^n$, the special subvarieties are, by definition, irreducible components of sets cut out by finitely many classical modular polynomial equations $\Phi_N(x_h,x_k)=0$ — where $\Phi_N$ is the modular polynomial relating $j(z)$ and $j(Nz)$ — together with equations $x_\ell=c$ with $c$ a singular modulus; allowing an arbitrary complex constant in the second type of equation defines the larger class of weakly special subvarieties. A component $A$ of an intersection $W\cap T$ of a subvariety $W\subset Y(1)^n$ with a special subvariety $T$ is called atypical (an unlikely intersection) when $\dim A>\dim W+\dim T-n$, i.e. when it is larger than dimension counting predicts. The Zilber–Pink conjecture, a general conjecture on unlikely intersections that contains the André–Oort, Manin–Mumford, and Mordell–Lang conjectures as special cases, predicts in this setting that every $W\subset Y(1)^n$ contains only finitely many maximal atypical subvarieties.

Ax–Schanuel-type theorems are the functional-transcendence input to such questions: they assert, roughly, that atypical algebraic relations among values of a transcendental function and its derivatives must be explained by the symmetry group of the function. Pila and Tsimerman (Duke Math. J. 2016) proved such a theorem for the $j$-function, both for $j$ alone and for $j$ together with its first two derivatives, the latter formulated on jet spaces $J^kM$ (the bundle of $k$-th order jets of holomorphic arcs in a complex manifold $M$, so that $J^1M$ is the tangent bundle). As an application they proved (their Theorem 7.1) that every $W\subset Y(1)^n$ contains only finitely many maximal strongly atypical subvarieties — atypical subvarieties on which no coordinate of $Y(1)^n$ is constant. The same paper posed two conjectures reaching beyond the modular curve: Conjecture 1.1, the Ax–Schanuel statement for the classical uniformization $\pi:\mathbb{H}_g\to\mathcal{A}_g$, in which the Siegel upper half-space $\mathbb{H}_g$ (the higher-dimensional analogue of $\mathbb{H}$) uniformizes $\mathcal{A}_g$, the moduli space of principally polarized abelian varieties — an abelian variety being a projective algebraic variety carrying a commutative group law, and a principal polarization a compatible choice of ample line bundle — and Conjecture 1.2, a with-derivatives (jet-space) version for the uniformization of an arbitrary Shimura variety, a Shimura variety being a quotient of a Hermitian symmetric domain (a complex manifold, such as $\mathbb{H}$, on which a real Lie group acts transitively by holomorphic isometries) by an arithmetic group, $Y(1)$ and $\mathcal{A}_g$ being the first examples. Their proof used the group-theoretic structure of $\mathrm{SL}_2(\mathbb{R})^n$ essentially, and the paper formulated these conjectures as the expected generalizations.

Both companion conjectures were subsequently resolved: Mok, Pila, and Tsimerman proved Ax–Schanuel for arbitrary pure Shimura varieties (their Theorem 1.1) together with the jet-space with-derivatives version (their Theorem 9.1), and the statement has since been extended to the universal abelian varieties and mixed Shimura varieties of Kuga type (families of abelian varieties over Shimura bases) by Gao, to all admissible variations of mixed Hodge structures (a Hodge structure being the linear-algebra datum recording how the complex cohomology of an algebraic variety decomposes, and a variation of them such data varying over a complex base) by Gao and Klingler, and to derivatives of mixed period mappings by Chiu. For Zilber–Pink itself, Habegger and Pila showed that the conjecture for products of modular curves follows from two ingredients: a modular Ax–Schanuel statement (then conjectural, and now more than fully available) and a lower bound for Galois orbits of the relevant unlikely-intersection points (the orbit of a point under the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/E)$ of a number field $E$ of definition). The generic part of the problem has since been completed: Aslanyan, Eterović, and Fowler settled the conjecture, in Pink's unlikely-intersections formulation, for all subvarieties of $Y(1)^n$ no projection of which to $\dim W+2$ coordinates is defined over $\overline{\mathbb{Q}}$, and Klingler and Tayou proved the analogous statement for subvarieties of mixed Shimura varieties not defined over $\overline{\mathbb{Q}}$ in a strong sense. What remains is the arithmetic case — subvarieties with $\overline{\mathbb{Q}}$-defined coordinate projections, including all subvarieties defined over number fields — where the missing ingredient is a large Galois orbits statement for optimal points of $W$ (subvarieties minimal for the defect $\delta(A)=\dim\langle A\rangle-\dim A$, where $\langle A\rangle$ is the smallest special subvariety containing $A$), which Daw's 2025 survey describes as 'still wide open, and, seemingly, very difficult'.

Problem Statement

Prove the Zilber–Pink conjecture for powers of the modular curve (Conjecture 7.1 of Pila–Tsimerman, Duke Math. J. 2016): for every integer $n\geq 1$ and every irreducible complex subvariety $W\subset Y(1)^n$, the set of maximal atypical subvarieties of $W$ is finite — maximal meaning maximal under inclusion among the atypical subvarieties of $W$ — where a subvariety $A\subset W$ is atypical if it is an irreducible component of $W\cap T$ for a special subvariety $T\subset Y(1)^n$ (special subvarieties being the irreducible components of sets defined by finitely many modular polynomial equations $\Phi_N(x_h,x_k)=0$ and equations $x_\ell=c$ with $c$ a singular modulus) with $\dim A>\dim W+\dim T-n$. Equivalently, prove that $W$ contains only finitely many optimal subvarieties, where $A\subset W$ is optimal if $\delta(B)<\delta(A)$ for every $B$ with $A\subsetneq B\subset W$, with defect $\delta(A)=\dim\langle A\rangle-\dim A$ and $\langle A\rangle$ the smallest special subvariety containing $A$. Theorem 7.1 of the source paper settles the strongly atypical case, in which no coordinate of $Y(1)^n$ is constant on $A$; the problem is to remove this restriction, that is, to handle atypical subvarieties on which some coordinates are constant and hence equal to singular moduli. In view of the 2025 results of Aslanyan–Eterović–Fowler, who settled in Pink's unlikely-intersections formulation all subvarieties of $Y(1)^n$ no projection of which to $\dim W+2$ coordinates is defined over $\overline{\mathbb{Q}}$, and of Klingler–Tayou, who settled the analogous statement for subvarieties of mixed Shimura varieties not defined over $\overline{\mathbb{Q}}$ in a strong sense, the open instances of the conjecture lie in the arithmetic case: subvarieties some projection of which to $\dim W+2$ coordinates is defined over $\overline{\mathbb{Q}}$, in particular all subvarieties defined over number fields.

References

1. Jonathan Pila and Jacob Tsimerman, 'Ax–Schanuel for the j-function', Duke Mathematical Journal 165, no. 13 (2016), 2587–2605, DOI 10.1215/00127094-3620005, https://doi.org/10.1215/00127094-3620005 (preprint arXiv:1412.8255, https://arxiv.org/abs/1412.8255)
2. Ngaiming Mok, Jonathan Pila, and Jacob Tsimerman, 'Ax-Schanuel for Shimura varieties', Annals of Mathematics (2) 189, no. 3 (2019), 945–978, DOI 10.4007/annals.2019.189.3.7, https://doi.org/10.4007/annals.2019.189.3.7 (preprint arXiv:1711.02189, https://arxiv.org/abs/1711.02189)
3. Philipp Habegger and Jonathan Pila, 'O-minimality and certain atypical intersections', Annales scientifiques de l'École normale supérieure (4) 49, no. 4 (2016), 813–858, DOI 10.24033/asens.2296, https://doi.org/10.24033/asens.2296 (preprint arXiv:1409.0771, https://arxiv.org/abs/1409.0771)
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