Let $\pi:\tilde{C}\rightarrow C$ be a ramified cover between two smooth curves. And consider a group scheme $\mathcal G$ over $\tilde{C}$, I have found two definitions for Weil restriction:

- $Res_{\tilde{C}/C} \mathcal G$ is the group scheme whose sheaf of section is $\pi_*\mathcal G$.
- The usual definition given in the book
*Néron Models*by Bosch, Luetkebohmert and Raynaud.

My questions:

- Are these two definitions equivalent? (I didn't understand exactly the first one!)
- How could one calculate explicitly the Weil restriction for the constant group scheme $G\times\tilde{C}$, for a usual group scheme $G$.

Is there any good reference? Thanks.