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arXiv · 2606.29369

Andr\'{e}'s theorem and weakly bounded height

Guy Fowler
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Abstract

Let $V \subset \mathbb{A}^2(\mathbb{C})$ be an algebraic curve such that $\mathrm{deg} X \neq \mathrm{deg} Y$, 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 $\max \{h(x), h(y)\} \leq c$ for every $(x, y) \in V$ with $x$ and $y$ both CM $j$-invariants. This establishes, for such curves, an effective version of the Andr\'{e}--Oort conjecture that has a better dependence on the height of $V$ than previous effective results.

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Guy Fowler. 2026-06-28. Andr\'{e}'s theorem and weakly bounded height. https://arxiv.org/abs/2606.29369

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