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 $\mathbb{E}{\otimes }_{\overline{\mathbb{Q}}}\mathbb{C}\to X_0{\times }_{\mathrm{Spec}\overline{\mathbb{Q}}}\mathrm{Spec}\mathbb{C}$ to $\mathbb{C}$ to be the base change of some connection on $\mathbb{E}$. A similar criterion is given for Higgs fields on $\mathbb{E}{\otimes }_{\overline{\mathbb{Q}}}\mathbb{C}$.

Final version