A Belyi-type criterion for vector bundles on curves defined over a number field

TL;DR

The paper proves a Belyi-type descent criterion for vector bundles on curves defined over a number field. By fixing a Belyi map $f_0: X_0 \to \mathbb{P}^1_{\overline{\mathbb{Q}}}$ with branch locus in $\{0,1,\infty\}$ and base-changing to $\mathbb{C}$, the direct image $f_*E$ splits as a sum of line bundles and acquires a parabolic structure; $E$ descends to $X_0$ if and only if $f_*E$ is isomorphic to a direct sum of line bundles with a parabolic structure defined over $\overline{\mathbb{Q}}$. The argument develops the parabolic direct-image theory, uses a Galois closure to relate pulls-back parabolic data to a descent, and proves that the existence of a $\overline{\mathbb{Q}}$-defined parabolic structure on $f_*E$ implies the existence of a descent of $E$ to $X_0$. The results connect arithmetic descent to the geometry of parabolic bundles and provide a practical criterion for when a complex vector bundle on a curve arises as the base change of a bundle defined over a number field.

Abstract

Let $X_0$ be an irreducible smooth projective curve defined over $\overline{\mathbb Q}$ and $f_0 : X_0 \rightarrow \mathbb{P}^1_{\overline{\mathbb Q}}$ a nonconstant morphism whose branch locus is contained in the subset $\{0,1, \infty\} \subset \mathbb{P}^1_{\overline{\mathbb Q}}$. For any vector bundle $E$ on $X = X_0\times_{{\rm Spec}\,\overline{\mathbb Q}} {\rm Spec} \mathbb{C}$, consider the direct image $f_*E$ on $\mathbb{P}^1_{\mathbb C}$, where $f= (f_0)_{\mathbb C}$. It decomposes into a direct sum of line bundles and also it has a natural parabolic structure. We prove that $E$ is the base change, to $\mathbb C$, of a vector bundle on $X_0$ if and only if there is an isomorphism $f_*E \stackrel{\sim}{\rightarrow} \bigoplus_{i=1}^r {\mathcal O}_{{\mathbb P}^1_{\mathbb C}}(m_i)$, where $r = {\rm rank}(f_*E)$, that takes the parabolic structure on $f_*E$ to a parabolic structure on $\bigoplus_{i=1}^r {\mathcal O}_{{\mathbb P}^1_{\mathbb C}}(m_i)$ defined over $\overline{\mathbb Q}$.

Theorems & Definitions (8)

• Theorem 1.1
• Proposition 2.1
• proof
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Remark 3.3