Timeline for How to get the dimension of Atkin-Lehner eigenspace or do you have any data already obtained?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 21, 2022 at 7:05 | comment | added | Kimball | Glad you got it sorted out! | |
| Sep 21, 2022 at 6:17 | comment | added | kslhg | @Kimball I had to delete the previous comment, in case someone else sees it because there was a serious error... And thanks to your comment, I also found the reason why the dimension I set up using newform decomposition was wrong (even though it was exactly the same way you said). I think I made a wrong formula of $\dim S^{+}(N)$ because I made mistake with genus of $X^{+}(N)$. Thanks to you, I fixed it, and thanks to you, I got the desired result! thanks. | |
| Sep 21, 2022 at 6:15 | vote | accept | kslhg | ||
| Sep 21, 2022 at 11:22 | |||||
| Sep 21, 2022 at 6:14 | vote | accept | kslhg | ||
| Sep 21, 2022 at 6:14 | |||||
| Sep 21, 2022 at 5:15 | comment | added | Kimball | @kslhg What you need to know is that if $M$ is coprime to $p$, then each newform in $S_k(M)$ gives one oldform in $S_k(pM)$ with $W_p$ eigenvalue $+1$ and similarly for $-1$. This should be written down somewhere in the original Atkin-Lehner paper, where I believe they explicitly construct the lifts with these eigenvalues. In your example, this means you get $dim S_8^{--}(22) = dim S_8^{new,--}(22) + dim S_8^{-}(11) = 4$, as the other 2 oldspaces contribute nothing. This matches what David Loeffler's code gives me. | |
| Sep 21, 2022 at 2:44 | comment | added | Kimball | @kslhg I edited my answer to indicate two ways you can treat the case of full cusp spaces, following what I did for newspaces. | |
| Sep 21, 2022 at 2:42 | history | edited | Kimball | CC BY-SA 4.0 |
added some clarifications about relation between new and oldspaces
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| Sep 21, 2022 at 2:39 | comment | added | Kimball | @kslhg Maybe it's helpful to see how the computation goes first for $N=p$, then $N=pq$ for distinct primes. For $N=p$, the dimension will be the Atkin-Lehner dimension of the newspace, plus the dimension in full level. For $N=pq$, say for the $(++)$ space, you want the new $(++)$ dimension, plus the $+$-space dimensions for newforms of levels $N_1 = p$ and $N_1 = q$, plus the level 1 dimension. | |
| Sep 20, 2022 at 23:42 | comment | added | kslhg | Thank you for your answer. I already knew about your work, but it was a study about new subspace, and what I wanted was the case including old subspace, so it wasn't the answer I wanted. It seems that exact dimension formulae including old one have not been made yet. I need dimensions for other reasons, but anyways, I recently made some conjecture to get dimensions, and I tried indirect calculations to verify them, but all failed. However, looking at your answer, I think something can be obtained by using the trace formula. I'll do some more study. Thank you again for your kind answer. | |
| Sep 20, 2022 at 13:06 | comment | added | David Loeffler | That's nice, I didn't know about this work of yours! | |
| Sep 20, 2022 at 12:46 | history | answered | Kimball | CC BY-SA 4.0 |