Holomorphic maps sharing preimages over finitely generated fields

Abstract

Let $R$ be a compact Riemann surface, and let $P:R\to {\mathbb{P}}^1(\mathbb{C})$ and $Q:R\to {\mathbb{P}}^1(\mathbb{C})$ be holomorphic maps. In this paper, we investigate the following problem: under what conditions do the preimages $P^{-1}(K)$ and $Q^{-1}(K)$ coincide for some infinite set $K$ contained in ${\mathbb{P}}^1(k)$, where $k$ is a finitely generated subfield of $\mathbb{C}$ (e.g., a number field)? Equivalently, we study holomorphic correspondences that admit an infinite completely invariant set contained in ${\mathbb{P}}^1(k)$. We show that if such a set exists, then there is a holomorphic Galois covering $\hat{I}˜:R_0\to {\mathbb{P}}^1(\mathbb{C})$, where $R_0$ has genus zero or one, such that $P$ and $Q$ are ``compositional left factors" of $\hat{I}˜.$ We also consider a more general equation $P^{-1}(K_1)=Q^{-1}(K_2),$ where $K_1$ and $K_2$ are infinite subsets of ${\mathbb{P}}^1(k)$.