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:

  1. $Res_{\tilde{C}/C} \mathcal G$ is the group scheme whose sheaf of section is $\pi_*\mathcal G$.
  2. 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.

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    $\begingroup$ The definitions look the same to me. $\endgroup$ – anon Apr 23 '15 at 1:34
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    $\begingroup$ In definition (1), $\pi_* \mathbb{G}$ is to be considered as a sheaf on the big Zariski (or etale, or ...) site of $C$, associating to a $C$-scheme $D$ the set ${\rm Hom}_{\tilde C}(D\times_C {\tilde C}, \mathcal{G})$, and then the definitions are equivalent. If we consider the small Zariski site of $C$, then of course the two definitions are not equivalent (often $\mathcal{G}$ will have no sections over Zariski opens). $\endgroup$ – Piotr Achinger Apr 23 '15 at 5:47
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    $\begingroup$ Calculating the Weil restriction of a constant group scheme should be similar to calculating the pushforward of the structure sheaf $\pi_* \mathcal{O}_C$. You should get some complicated answer in terms of things like the ramification data of the cover. A simpler thing to understand is the generic fibre of this, which is just the Weil restriction of a constant group scheme with respect to a quadratic field extension. $\endgroup$ – Daniel Loughran Apr 23 '15 at 7:56
  • $\begingroup$ Thanks for your really helpful answers, @Daniel Loughran I think your comment need to be expended to a good answer!! could you please do it? I am little bit confused and I don't see what you said... $\endgroup$ – Z.A.Z.Z Apr 23 '15 at 8:07
  • $\begingroup$ Taking $G=GL_r(\mathcal O_\tilde{C})$ for example, is it true that $Res_{\tilde{C}/C}(G\times \tilde{C})=GL_r(\pi_*\mathcal O_\tilde{C})$ ? $\endgroup$ – Z.A.Z.Z Apr 23 '15 at 14:36

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