Timeline for isogeny and congruence subgroup
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5, 2013 at 15:29 | vote | accept | Lan | ||
| Dec 1, 2013 at 7:02 | comment | added | user76758 | To clarify the end of Jim's answer, for a number field $F$ and $n\ge 2$ there are many finite-index non-congruence subgroups $\Gamma_2\subset {\rm{PGL}}_n(O_F)$, yet the cokernel ${\rm{H}}^1(O_F,\mu_n)$ of $f:{\rm{SL}}_n(O_F) \rightarrow {\rm{PGL}}_n(O_F)$ is finite, so replacing $\Gamma_2$ with a finite-index subgroup ensures $\Gamma_2=f(\Gamma_1)$ for $\Gamma_1=f^{-1}(\Gamma_2)$. But $\Gamma_1$ has finite index in ${\rm{SL}}_n(O_F)$, so it is a congruence subgroup if $n \ge 3$ and $F$ has a real place. Then $\Gamma_1$ is congruence with image containing no congruence subgroup. | |
| Nov 30, 2013 at 12:35 | comment | added | Lan | I am sorry for making mistake. I edit hte queston. I am especially concern the example I added. | |
| Nov 30, 2013 at 2:57 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
added 35 characters in body
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| Nov 30, 2013 at 1:39 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
added 52 characters in body
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| Nov 30, 2013 at 1:03 | history | answered | Jim Humphreys | CC BY-SA 3.0 |