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

TL;DR

This paper establishes uniform effective bounds for sums of powers of singular moduli in complex n-space, extending Andre9--Oort type results with explicit constants and classifications.

Contribution

It proves an effective, uniform Andre9--Oort-type theorem for sums of powers of singular moduli, including explicit bounds and complete classifications in specific cases.

Findings

Existence of an effective constant c(m, n) bounding discriminants

Complete classification of singular moduli triples with rational linear combinations

Extension of Andre9--Oort results to sums of powers in c4^n

Abstract

We prove an Andr\'e--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 $\Delta_1, \ldots, \Delta_n$ 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 $\# \{ \Delta_i : \mathbb{Q}(\sqrt{\Delta_i}) = K_*\} \leq 1$, then $\max_i \lvert \Delta_i \rvert \leq c(m, n)$. 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 $a_1, a_2, a_3 \in \mathbb{Q} \setminus \{0\}$.