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

Abstract

We prove an André--Oort-type result for a family of hypersurfaces in ${\mathbb{C}}^n$ that is both uniform and effective. Let $K_{\ast }$ 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 $x_1,\dots ,x_n$ with respective discriminants ${\Delta }_1,\dots ,{\Delta }_n$ are such that $a_1{x}_1^m+\dots +a_n{x}_n^m\in \mathbb{Q}$ for some $a_1,\dots ,a_n\in \mathbb{Q}\setminus \{0\}$ and $\#\{{\Delta }_i:\mathbb{Q}(\sqrt{{\Delta }_i})=K_{\ast }\}\le 1$, then ${\max }_i|{\Delta }_i|\le c(m,n)$. 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 $a_1,a_2,a_3\in \mathbb{Q}\setminus \{0\}$.

32 pages, to appear in Math. Proc. Camb. Philos. Soc

References in corpus (1)

• Equations in three singular moduli: the equal exponent case