Timeline for Understanding moduli of shtukas of non-minuscule cocharacter
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 8, 2020 at 22:40 | comment | added | xir | oh i see, very neat. i'm satisfied; thank you! | |
| May 8, 2020 at 22:38 | comment | added | dorebell | I can't find a reference right now but the argument is: The category of $G$-torsors is the same thing as the category of faithful exact tensor functors from the category of representations of $G$ into vector bundles. So a $G$-shtuka is the same thing as a collection of $GL(V)$-shtukas for each representation of $V$ which are compatible with tensor products etc. Thus if I had a $G$-shtuka with one leg, all of the corresponding $\GL(V)$ shtukas would be trivial, and thus the original $G$-shtuka must be as well. | |
| May 8, 2020 at 22:16 | comment | added | xir | as i said, i agree for GLn. i'm not sure i understand the second part, though i think i get why my example above doesn't work; do you have a reference? | |
| May 8, 2020 at 18:48 | comment | added | dorebell | For $\mathrm{GL}_n$, this should come from the fact that pulling back a vector bundle $\mathscr{E}$ on $X_S$ by $F = 1 \times \mathrm{Frob}_S$ doesn't change the degree, so an everywhere-defined morphism $F^*\mathscr{E} \rightarrow \mathscr{E}$ that only vanishes at one point must be an isomorphism. For general groups, I think you can just use the above argument via the Tannakian description of torsors. | |
| May 8, 2020 at 15:59 | comment | added | xir | maybe you mean for GLn? i can see that, at least (reduce to the case of line bundles via determinants and then it's just basic facts about positivity), but it seems like there are shtukas with one leg for, say, PGLn. take for instance simply the natural ideal sheaf inclusion O(-p)^n -> O^n for some rational point p; this is a shtuka for PGLn, no? | |
| May 7, 2020 at 14:27 | comment | added | xir | thanks very much for the detailed response! one more naive question: could you elaborate why there are "no shtukas with one leg"? in what setting is this true? | |
| May 7, 2020 at 14:17 | vote | accept | xir | ||
| May 7, 2020 at 3:53 | history | answered | dorebell | CC BY-SA 4.0 |