Timeline for rationality question while dealing with an isogeny
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 3, 2014 at 23:36 | vote | accept | thierry stulemeijer | ||
| Mar 3, 2014 at 23:35 | comment | added | thierry stulemeijer | Indeed, and again very interesting, thanks ! As for my description, it is equivalent to yours under an obvious base change, and just comes from looking at the isogeny at level of coordinate rings, restricting it to torus and then "choosing bad coordinates". | |
| Mar 3, 2014 at 23:12 | comment | added | user76758 | @thierrystulemeijer: On character groups the map is an inclusion $\overline{L} \rightarrow L$ between finite free $\mathbf{Z}$-modules of rank $p-1$ such that the cokernel is cyclic of order $p$, so in suitable bases it looks like the direct product of the identity map on $\mathbf{Z}^{p-2}$ and the index-$p$ inclusion of $\mathbf{Z}$ into itself via multiplication by $\mathbf{Z}$. Now apply ${\rm{Hom}}(\cdot, {\rm{GL}}_1)$ to turn that back into a map of tori. So all I am doing is "choosing good coordinates" for describing the map on tori. I don't understand your proposed description, sorry. | |
| Mar 3, 2014 at 23:07 | comment | added | thierry stulemeijer | That is great ! Just to be sure I understand properly : I don't see why your retsricted morphism on torus is the identity for $p-2$ components. Instead, I think the formula is something like $\mathbb{F}_{p}[\overline{D}]=\mathbb{F}_{p}[X_{1},...,X_{p}]_{1/det} \rightarrow \mathbb{F}_{p}[D]=\mathbb{F}_{p}[Y_{1},...,Y_{p}]_{1/det} : X_{i}\mapsto Y_{1}^{p-1}Y_{i} $. Also, I will edit the question to take into account your last paragraph. | |
| Mar 3, 2014 at 14:24 | history | answered | user76758 | CC BY-SA 3.0 |