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I'm stucking on the proof of the Lemma 3.12 of A p-adic analogue of Borel’s theorem.

Here $\mathcal A_{g,\mathrm K}$ is just a shimura variety defined over $\mathbb Z_p$, and full level $\ell$ structure means the $\ell$-part of $\mathrm K$ is contained in a principal congruence subgroup of $G(\mathbb Z_\ell)$. Knowing these there will be no problem for reading the proof.

I have the following questions:

  1. Since the abelian scheme is pulled back to $G_m$, what does he mean by its reduction at $0$? Does it mean by extending this abelian scheme to $0$? But then, how?

  2. In the second sentence of the second paragraph, what does "toric monodromy" mean?

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  • $\begingroup$ Re. 1., the local ring of $A^1$ at 0 is a DVR, so an abelian scheme over its fraction field will extend canonically to a smooth separated group scheme - the Neron model - over the whole DVR. One can take the fiber of this at 0 - such a fiber is generally a disconnected group scheme whose connected component is a semiabelian variety. The authors seem to be claiming that this fiber is connected, but I could not see why this is true in their situation. Generally I could not follow the proof of Lemma 3.12 at all... $\endgroup$
    – Satan's Minion
    Commented Mar 12 at 11:28
  • $\begingroup$ @Richard: Regarding (2), the authors are using the Neron-Ogg-Shafarevich criterion, which describes the reduction of an abelian scheme in terms of the local monodromy on it's $\ell$-adic Tate module. By "toric" I think they just mean "lies in a torus," i.e. since they in the tame setting, this just means that a generator of inertia goes to a matrix which is diagonalizable over $\overline{\mathbb{Q}_\ell}$. $\endgroup$
    – Daniel Litt
    Commented Mar 13 at 0:59
  • $\begingroup$ @Satan'sMinion Thanks, but why is the connected component semiabelian? $\endgroup$
    – Richard
    Commented Mar 13 at 12:34

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