Some uniform effective results on André--Oort for sums of powers in $\mathbb{C}^n$

TL;DR

This work advances effective André–Oort-type statements for sums of powers of singular moduli in $\mathbb{C}^n$ by isolating an exceptional imaginary quadratic field $K_*$ and combining Pila’s uniform André–Oort framework with Tatuzawa-style class-number refinements. It proves an effective bound $c(m,n)$ under a controlled occurrence of discriminants and exhibits a uniform, explicit version under stronger discriminant-structure assumptions, with a completely explicit result for the case $ (m,n)=(1,3)$. Central to the approach are precise analyses of singular moduli with the same discriminant, and fields generated by linear combinations of powers, complemented by auxiliary results on the size and arithmetic of discriminants and class numbers. The paper also provides a fully explicit classification of all triples of singular moduli $(x,y,z)$ with a rational linear relation $Ax+By+Cz$, thereby yielding concrete arithmetic constraints on these special points and their fields. Overall, the results enhance both the effectiveness and the uniformity of André–Oort-type statements in a natural, nontrivial family of hypersurfaces in $\mathbb{C}^n$, with potential for algorithmic and computational applications in the arithmetic of CM points.

Abstract

We prove an André--Oort-type result for a family of hypersurfaces in $\mathbb{C}^n$ that is both uniform and effective. Let $K_*$ denote the single exceptional imaginary quadratic field which occurs in the Siegel--Tatuzawa lower bound for the class number. We prove that, for $m, n \in \mathbb{Z}_{>0}$, there exists an effective constant $c(m, n)>0$ with the following property: if pairwise distinct singular moduli $x_1, \ldots, x_n$ with respective discriminants $Δ_1, \ldots, Δ_n$ are such that $a_1 x_1^m + \ldots + a_n x_n^m \in \mathbb{Q}$ for some $a_1, \ldots, a_n \in \mathbb{Q} \setminus \{0\}$ and $\# \{ Δ_i : \mathbb{Q}(\sqrt{Δ_i}) = K_*\} \leq 1$, then $\max_i \lvert Δ_i \rvert \leq c(m, n)$. In addition, we prove an unconditional and completely explicit version of this result when $(m, n) = (1, 3)$ and thereby determine all the triples $(x_1, x_2, x_3)$ of singular moduli such that $a_1 x_1 + a_2 x_2 + a_3 x_3 \in \mathbb{Q}$ for some $a_1, a_2, a_3 \in \mathbb{Q} \setminus \{0\}$.

Theorems & Definitions (67)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Proposition 2.2
• proof
• Lemma 2.3