Connections and Higgs bundles on curves defined over a number field

Abstract

Let $X_0$ be an irreducible smooth projective curve defined over $\overline{\mathbb Q}$ and $\mathbb E$ a vector bundle on $X_0$. We give a criterion for connections on the base change ${\mathbb E}\otimes_{\overline{\mathbb Q}}{\mathbb C} \rightarrow X_0\times_{{\rm Spec}\,\overline{\mathbb Q}} {\rm Spec}\,\mathbb{C} $ to $\mathbb C$ to be the base change of some connection on $\mathbb E$. A similar criterion is given for Higgs fields on ${\mathbb E}\otimes_{\overline{\mathbb Q}}{\mathbb C}$.