Timeline for Some questions about $\ell$-adic monodromy
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Mar 13 at 12:34 | comment | added | Richard | @Satan'sMinion Thanks, but why is the connected component semiabelian? | |
| Mar 13 at 0:59 | comment | added | Daniel Litt | @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 $\overline{\mathbb{Q}_\ell}$. | |
| Mar 12 at 11:28 | comment | added | Satan's Minion | 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... | |
| Mar 12 at 7:30 | history | asked | Richard | CC BY-SA 4.0 |