A map between moduli spaces of connections

Let $C$ be an elliptic curve over $\mathbb{C}$, and $\pi \colon C \to \mathbb{P}^1$ a ramified double cover. In his thesis (done here at the IRMAR), Nestor Fernández Vargas described a procedure to transform parabolic bundles over $\mathbb{P}^1$ into parabolic bundles over $C$, using the map $\pi$. This induces a morphism between the moduli spaces of parabolic bundles.

In this talk I'm going to present how to extend the former to a map between moduli spaces of parabolic connections (and describe the moduli spaces involved).