Timeline for Voronoi formula on $\mathrm{GL}_4$ in the level aspect with ramification
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| Dec 5 at 8:06 | comment | added | hofnumber | @WindomEarle Great thanks for help. Yes, you are right. Noting that Edgar Assing and Abhishek Saha have referred what these are for the $GL_2$ case (see, e.g., Page 9 of this paper--Yet another GL2 subconvexity result). But, for more general $GL_n$ case, there does not seem to be a clear record in the reference. | |
| Dec 5 at 6:51 | comment | added | Windom Earle | Up to scaling these are the Dirichlet coefficients of the local L-factor at p. This can be computed reasonably explicitly in the relevant case. | |
| Dec 5 at 3:22 | comment | added | hofnumber | @WindomEarle Dear professor, sorry to disturb, I want to ask you whether or not we have the exact value for the local Whittaker functions at ramified place $p$ ? That is, when $p$ dividing the level $N$, say, of the form $F$, what looks like for $$W_{F,p}\left(\left(\begin{matrix}p^k& & &\\ &1& &\\ &&1&\\ &&&1 \end{matrix}\right)\right)$$ for positive integer $k$? In other word, how about the Fourier coefficient $A_F(p^k,1,1,1)$ for $p|N$, upon recalling the relation between Fourier coefficients and Whittaker functions as shown in Definition 5.1 of Corbett's paper. | |
| Dec 4 at 9:51 | comment | added | hofnumber | @WindomEarle Dear professor, many thanks for explanation. | |
| Nov 30 at 9:06 | comment | added | Windom Earle | Computing the local transforms in Corbett's formula is not a problem. These give indeed the desired hyper Kloosterman sums. But arranging them correctly requires playing around with the Hecke relations at the (ramified place) p. | |
| Nov 28 at 2:26 | history | edited | hofnumber | CC BY-SA 4.0 |
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| Nov 28 at 1:54 | comment | added | hofnumber | @WindomEarle Dear professor, I want to consult you again: in the case of $p|c$, if the Voronoi formula should be the same as that for trivial level case (i.e., p=1)? Since according to Eqn. (1) of Zhou's paper mentioned above, even the well-known Voronoi in the ordinary $GL_2$ case, the level $p$ does not involve if $p|c$. In other word, I wanna ask if the form (1) in this post should be true? Many thanks. | |
| Nov 28 at 1:47 | history | edited | hofnumber | CC BY-SA 4.0 |
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| Nov 28 at 1:44 | comment | added | hofnumber | @WindomEarle Great thanks for kindly guide, dear professor. Very deep thanks to you for telling me how to apply Corbett's formula to work out the symmetric-square lift case roughly two years ago in this post (mathoverflow.net/questions/337721/…). As mentioned by Prof. Peter Humphries, Corbett's formula can solve this case; however, to explicitly translate into the classical formula looks a bit tough for me now. | |
| Nov 27 at 14:51 | comment | added | Windom Earle | The assumption that f has prime level p greatly restricts the possibilities for the local component of the corresponding automorphic representation at p. This makes it relatively easy to work out how Corbett's formula looks explicitly in this case. | |
| Nov 27 at 0:28 | comment | added | hofnumber | @PeterHumphries Many thanks for pointing out this blur. | |
| Nov 27 at 0:18 | comment | added | Peter Humphries | I think there is a translation error on your behalf: if the level and the modulus are jointly ramified, then they are not coprime. | |
| Nov 26 at 23:21 | comment | added | hofnumber | @PeterHumphries Thanks, Peter, I will check this again. It seems that Corbett was more concerned about the co-prime situation, as it mentioned "...Of particular interest are those primes at which the level and modulus are jointly ramified..." in the Abstract. | |
| Nov 26 at 14:53 | comment | added | Peter Humphries | Corbett's paper deals with this particular situation (and much much more). Of course, one has to rewrite his adelic statement back into a classical statement. | |
| Nov 26 at 9:54 | history | edited | hofnumber | CC BY-SA 4.0 |
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| Nov 26 at 9:09 | history | edited | YCor | CC BY-SA 4.0 |
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| Nov 26 at 8:48 | history | edited | hofnumber | CC BY-SA 4.0 |
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| Nov 26 at 5:32 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
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| Nov 26 at 4:34 | history | edited | hofnumber | CC BY-SA 4.0 |
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| S Nov 26 at 4:33 | review | First questions | |||
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| S Nov 26 at 4:33 | history | asked | hofnumber | CC BY-SA 4.0 |