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

Abstract

Let $X_0$ be an irreducible smooth projective curve defined over $\overline{\mathbb{Q}}$ and $f_0:X_0\to {\mathbb{P}}_{\overline{\mathbb{Q}}}^1$ a nonconstant morphism whose branch locus is contained in the subset $\{0,1,\infty \}\subset {\mathbb{P}}_{\overline{\mathbb{Q}}}^1$. For any vector bundle $E$ on $X=X_0{\times }_{\mathrm{Spec}\overline{\mathbb{Q}}}\mathrm{Spec}\mathbb{C}$, consider the direct image $f_{\ast }E$ on ${\mathbb{P}}_{\mathbb{C}}^1$, 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 $f_{\ast }E\stackrel{\sim }{\to }{⨁}_{i=1}^r{\mathcal{O}}_{{\mathbb{P}}_{\mathbb{C}}^1}(m_i)$, where $r=\mathrm{rank}(f_{\ast }E)$, that takes the parabolic structure on $f_{\ast }E$ to a parabolic structure on ${⨁}_{i=1}^r{\mathcal{O}}_{{\mathbb{P}}_{\mathbb{C}}^1}(m_i)$ defined over $\overline{\mathbb{Q}}$.

Final version; to appear in Michigan Mathematical Journal