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 $f_0 : X_0 \rightarrow \mathbb{P}^1_{\overline{\mathbb Q}}$ 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 $\overline{\mathbb Q}$.