Multiplicatively dependent triples of singular moduli: the Bilu–Gun–Tron classification conjecture

Singular moduli — the values of the modular $j$-function at imaginary quadratic (CM) points — are classical objects of arithmetic geometry. Bilu, Gun and Tron proved that if distinct nonzero singular moduli $x,y,z$ and nonzero integers $m,n,r$ satisfy $x^m y^n z^r\in\mathbb{Q}^\times$, then the discriminants of $x,y,z$ are bounded by $10^{10}$. They describe three structural families of such triples (rational, quadratic, and cubic type) and conjecture that, up to permuting $x,y,z$, these exhaust all examples; the equal-exponent case $m=n=r$ was settled earlier by Fowler, but genuinely unequal exponents remain open. The task is to settle the conjecture — prove exhaustiveness of the three families, or exhibit a triple outside them.

Background

A singular modulus is the value $j(\tau)$ of the classical modular $j$-function (the generator of the field of modular functions for the modular group $SL_2(\mathbb{Z})$, the group of $2\times 2$ integer matrices of determinant 1) at a point $\tau$ of the Poincaré upper half-plane satisfying a quadratic equation with integer coefficients. These are exactly the algebraic values of $j$: by the theory of complex multiplication (CM), a singular modulus is the $j$-invariant of an elliptic curve whose endomorphism ring is an order $\mathcal{O}_\Delta=\mathbb{Z}[(\Delta+\sqrt{\Delta})/2]$ of discriminant $\Delta<0$ in an imaginary quadratic field. A singular modulus $x$ is an algebraic integer of degree equal to the class number $h(\Delta_x)$ of its order — the number of equivalence classes of ideals of $\mathcal{O}_\Delta$ — and the $h(\Delta)$ singular moduli of a given discriminant form a single orbit under the Galois group of $\mathbb{Q}$: they are conjugates, namely the full root set of the Hilbert class polynomial $H_\Delta$, an explicit integer polynomial of degree $h(\Delta)$ whose roots are exactly these moduli. For example, the class-number-1 discriminants $-4,-7,-8,-11,-19,-43,-67,-163$ give rational singular moduli such as $j(i)=1728$; the discriminant $-3$ is excluded throughout because $j((1+\sqrt{-3})/2)=0$ is not invertible.

Algebraic numbers $x_1,\dots,x_k$ are called multiplicatively dependent when some nontrivial product of integral powers of them equals 1. Here the object of study is the variant $x^m y^n z^r\in\mathbb{Q}^\times$ (the multiplicative group of nonzero rationals) with $x,y,z$ distinct nonzero singular moduli and $m,n,r$ nonzero integers. Such questions are explicit instances of the unlikely-intersections philosophy of Diophantine geometry — the principle that a subvariety meets the 'special' points of a variety only for structural reasons, as in the André–Oort conjecture on special points — here for the modular curve $Y(1)$ (the affine $j$-line) together with the multiplicative group $\mathbb{G}_m$; they complement the classical theory of linear relations among singular moduli.

The prior state of the art is as follows. Pila and Tsimerman (2017) proved, non-effectively (the argument goes through Siegel's celebrated but ineffective lower bound for class numbers, which yields no explicit numerical bound), that for every $k$ only finitely many $k$-tuples of nonzero singular moduli are multiplicatively dependent while every proper subset is independent. Riffaut (2019) made this effective and fully explicit for $k=2$, classifying all instances of $x^m y^n\in\mathbb{Q}^\times$ for distinct singular moduli. For $k=3$, Fowler (2020, 2023) settled the equal-exponent case, classifying all triples with $x_1x_2x_3\in\mathbb{Q}^\times$ and, more generally, all solutions of $(x_1^{\pm1}x_2^{\pm1}x_3^{\pm1})^a\in\mathbb{Q}^\times$. Bilu, Gun and Tron (preprint 2022; published version Algebra & Number Theory 20 (2026)) then obtained the effective result for general exponents: $x^m y^n z^r\in\mathbb{Q}^\times$ implies $\max\{|\Delta_x|,|\Delta_y|,|\Delta_z|\}<10^{10}$. They also exhibit three families of examples — rational, quadratic, and cubic type — and conjecture that up to permuting $x,y,z$ there are no others; for instance the Pila–Tsimerman rational-type identity $(2^6 3^3)^{10}(-2^{15})^{6}(-2^{15}3^3)^{-10}=1$ involves the moduli of discriminants $-4,-11,-19$. The obstruction the authors identify is quantitative: the number of singular moduli with $|\Delta|\le B$ grows on the order of $B^{3/2}$, so the range below $10^{10}$ is far too large for brute-force enumeration of triples, and finishing the classification requires first sharpening the explicit bound — or finding a new structural argument.

Problem Statement

Settle the classification conjecture of Bilu–Gun–Tron and thereby complete the classification of multiplicative relations among triples of singular moduli.

Let $x,y,z$ be distinct nonzero singular moduli and $m,n,r$ nonzero integers such that

$$x^{m}y^{n}z^{r}\in\mathbb{Q}^{\times}.$$

Theorem 1.1 of Bilu–Gun–Tron gives $\max\{|\Delta_x|,|\Delta_y|,|\Delta_z|\}<10^{10}$, and all currently known examples belong, up to permuting $x,y,z$, to one of three families:

1. Rational type: $h(\Delta_x)=h(\Delta_y)=h(\Delta_z)=1$ with all three discriminants different from $-3$; then $x,y,z\in\mathbb{Q}^\times$ and any nonzero exponents work.
2. Quadratic type: $h(\Delta_x)=1$ with $\Delta_x\neq-3$, $\Delta_y=\Delta_z$, and $h(\Delta_y)=2$; then $x\in\mathbb{Q}^\times$ and $y,z$ are conjugate of degree 2, so $x^m y^n z^n\in\mathbb{Q}^\times$ for any nonzero $m,n$.
3. Cubic type: $\Delta_x=\Delta_y=\Delta_z$ with $h(\Delta_x)=3$; then $x,y,z$ are the three conjugates of a degree-3 modulus and $xyz\in\mathbb{Q}^\times$.

The problem is to prove that, up to permuting $x,y,z$, there are no other examples — which completes the classification — or else to exhibit a triple of distinct nonzero singular moduli outside these three families, together with explicit exponents $m,n,r$, such that $x^m y^n z^r\in\mathbb{Q}^\times$. The route identified by the authors is to sharpen the explicit numerical bound $10^{10}$ of Theorem 1.1 down to a range where the remaining finite verification is computationally feasible and to carry out that verification, but a solution by any method is admissible. A valid proof must handle genuinely unequal exponents: the equal-exponent case $m=n=r$ is already settled by Fowler, while the quadratic family realizes exponents $(m,n,n)$.

References

1. Yuri Bilu, Sanoli Gun, Emanuele Tron, 'Effective multiplicative independence of three singular moduli', Algebra & Number Theory 20 (2026), no. 6, 1073–1123. DOI: 10.2140/ant.2026.20.1073; arXiv:2207.05183. https://doi.org/10.2140/ant.2026.20.1073
2. Jonathan Pila, Jacob Tsimerman, 'Multiplicative relations among singular moduli', Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5) 17 (2017), no. 4, 1357–1382. DOI: 10.2422/2036-2145.201502_001. https://doi.org/10.2422/2036-2145.201502_001
3. Antonin Riffaut, 'Equations with powers of singular moduli', International Journal of Number Theory 15 (2019), no. 3, 445–468. DOI: 10.1142/S1793042119500234. https://doi.org/10.1142/S1793042119500234
4. Guy Fowler, 'Triples of singular moduli with rational product', International Journal of Number Theory 16 (2020), no. 10, 2149–2166. DOI: 10.1142/S1793042120501110. https://doi.org/10.1142/S1793042120501110
5. Guy Fowler, 'Equations in three singular moduli: the equal exponent case', Journal of Number Theory 243 (2023), 256–297. DOI: 10.1016/j.jnt.2022.09.012. https://doi.org/10.1016/j.jnt.2022.09.012
6. Guy Fowler, 'Multiplicative independence of modular functions', Journal de Théorie des Nombres de Bordeaux 33 (2021), no. 2, 459–509. DOI: 10.5802/jtnb.1167. https://doi.org/10.5802/jtnb.1167
7. Vahagn Aslanyan, Sebastian Eterović, Guy Fowler, 'Multiplicative relations among differences of singular moduli', Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (published online December 2024). DOI: 10.2422/2036-2145.202309_020; arXiv:2308.12244. https://doi.org/10.2422/2036-2145.202309_020
8. Guy Fowler, 'Some uniform effective results on André–Oort for sums of powers in $\mathbb{C}^n$', Mathematical Proceedings of the Cambridge Philosophical Society 180 (2026), no. 3, 607–641. DOI: 10.1017/S0305004125101825; arXiv:2405.06456. https://doi.org/10.1017/S0305004125101825
9. Guy Fowler, Emanuele Tron, 'Distinct differences of singular moduli', Manuscripta Mathematica 177 (2026), no. 5 (published online August 2026). DOI: 10.1007/s00229-026-01745-z; arXiv:2503.19686. https://doi.org/10.1007/s00229-026-01745-z
10. Yuri Bilu, Florian Luca, 'Multiplicative dependence in the sumset of multiplicative groups', arXiv:2607.28857 (2026). https://arxiv.org/abs/2607.28857