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Holomorphic maps sharing preimages over finitely generated fields

Fedor Pakovich

TL;DR

The paper addresses when holomorphic maps $P,Q:R\to\mathbb{P}^1(\mathbb{C})$ share infinite preimage sets over a finitely generated field. By translating the problem into holomorphic correspondences and fiber-product geometry, it proves that such sharing forces a low-genus holomorphic Galois cover $\Theta:R_0\to\mathbb{P}^1(\mathbb{C})$ with $g(R_0)\le 1$, through which $P$ and $Q$ (and any related maps) factor: $\Theta=P\circ U=Q\circ V$. The results extend to the equal-set case $P^{-1}(K)=Q^{-1}(K)$ and to generalized $P^{-1}(K_1)=Q^{-1}(K_2)$, supported by fiber-product normalization, Faltings’ theorem, and orbifold-type considerations. The authors also provide explicit constructions of such maps and invariant preimage sets, illustrating the structural constraints and outlining when nontrivial right-factors or finite-rotation groups arise. Overall, the work gives a precise geometric framework linking preimage-sharing to genus-one (or lower) Galois factorizations in holomorphic dynamics on Riemann surfaces, with sets $K_i$ drawn from 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 $Θ: R_0 \to \mathbb P^1(\mathbb C)$, where $R_0$ has genus zero or one, such that $ P $ and $ Q $ are ``compositional left factors" of $ Θ.$ 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 $\mathbb P^1(k)$.

Holomorphic maps sharing preimages over finitely generated fields

TL;DR

The paper addresses when holomorphic maps share infinite preimage sets over a finitely generated field. By translating the problem into holomorphic correspondences and fiber-product geometry, it proves that such sharing forces a low-genus holomorphic Galois cover with , through which and (and any related maps) factor: . The results extend to the equal-set case and to generalized , supported by fiber-product normalization, Faltings’ theorem, and orbifold-type considerations. The authors also provide explicit constructions of such maps and invariant preimage sets, illustrating the structural constraints and outlining when nontrivial right-factors or finite-rotation groups arise. Overall, the work gives a precise geometric framework linking preimage-sharing to genus-one (or lower) Galois factorizations in holomorphic dynamics on Riemann surfaces, with sets drawn from finitely generated fields.

Abstract

Let be a compact Riemann surface, and let and be holomorphic maps. In this paper, we investigate the following problem: under what conditions do the preimages and coincide for some infinite set contained in , where is a finitely generated subfield of (e.g., a number field)? Equivalently, we study holomorphic correspondences that admit an infinite completely invariant set contained in . We show that if such a set exists, then there is a holomorphic Galois covering , where has genus zero or one, such that and are ``compositional left factors" of We also consider a more general equation where and are infinite subsets of .

Paper Structure

This paper contains 6 sections, 13 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Holomorphic maps sharing preimages of different sets
  3. Fiber products and normalizations
  4. Proof of Theorem \ref{['t0']} and its extensions
  5. Holomorphic maps sharing preimages of the same set
  6. Examples of holomorphic maps sharing preimages

Key Result

Theorem 1.1

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 having no non-trivial common compositional right factor. Assume that the equality holds for some infinite sets $K_1$ and $K_2$ contained in ${\mathbb P}^1({\bm k})$, where $\bm k$ is a finitely generated subfield of ${\mathbb C}$. Then, there exist comp

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 3.4
  • ...and 3 more