André's theorem and weakly bounded height

Abstract

Let $V\subset {\mathbb{A}}^2(\mathbb{C})$ be an algebraic curve such that $\mathrm{deg}X\ne \mathrm{deg}Y$, where $X,Y$ denote the coordinate functions on ${\mathbb{A}}^2(\mathbb{C})$ restricted to $V$. We prove there exists an effectively computable constant $c$, that depends linearly on the height of $V$, such that $\max \{h(x),h(y)\}\le c$ for every $(x,y)\in V$ with $x$ and $y$ both CM $j$-invariants. This establishes, for such curves, an effective version of the André--Oort conjecture that has a better dependence on the height of $V$ than previous effective results.