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

TL;DR

This paper establishes a criterion for when a vector bundle on a complex curve, defined over an algebraic closure of Q, descends from a curve over a number field, using a Belyi-type condition involving the direct image of the bundle.

Contribution

It provides a new criterion characterizing vector bundles over curves defined over number fields via their direct images and associated parabolic structures, extending Belyi's theorem to vector bundles.

Findings

Characterization of vector bundles via direct image decomposition

Identification of parabolic structures over algebraic numbers

Criterion for descent of vector bundles from complex to number fields

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 $\overline{\mathbb Q}$.