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hofnumber
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A question involving the three-dimensional Kloosterman sum
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A question involving the three-dimensional Kloosterman sum
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A question involving the three-dimensional Kloosterman sum
@WindomEarle Sorry, there is a typo; have re-edited. Thanks!
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A question involving the three-dimensional Kloosterman sum
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A question involving the three-dimensional Kloosterman sum
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A question involving the three-dimensional Kloosterman sum
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Whether or not the Maass form for $\Gamma _0(N)$ on $GL(3)$ covers the classical symmetric lift of a newform on $GL(2)$?
@PeterHumphries Many thanks!
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Whether or not the Maass form for $\Gamma _0(N)$ on $GL(3)$ covers the classical symmetric lift of a newform on $GL(2)$?
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Whether or not the Maass form for $\Gamma _0(N)$ on $GL(3)$ covers the classical symmetric lift of a newform on $GL(2)$?
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Voronoi formula for the symmetric $L$-function with level $N $
@WindomEarle You are right! However, I still don't understand if the definition of Maass form for the congruence subgroup $\Gamma_0(N^2)$ (see page 5 on the paper arxiv.org/pdf/1806.10786.pdf) given by Zhou covers the classical form $\text{sym}^2f$ of a typical $GL_2$ cusp form of level $N$?? If so, probably the formula (2) in Zhou's paper can be directly applied. Many thanks.
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Voronoi formula for the symmetric $L$-function with level $N $
...Also, I am not familiar with the representation theory so much. So, again, I post my puzzle on MO bothering some experts like you, wishing to get some help. I don't know how to say thanks for so much help from you, professor. Much obliged if some information can be given or some reference can be guided. Great great thanks again and again!!!
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Voronoi formula for the symmetric $L$-function with level $N $
@WindomEarle Dear professor, sorry to spend your valuable time answering some might naive questions with no any reward. The thing is that recently the referee in my draft pointed out that Zhou's formula (2) appears to be used, not even deducing the Voronoi formula again. But I feel that Zhou's formula might not directly applied to the $\text{sym}^2f$ with $f$ being a newform of prime level on $GL_2$...
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Voronoi formula for the symmetric $L$-function with level $N $
@WindomEarle Dear professor, I am also not clear that, regarding the Voronoi formula for symmetric lift on $GL_2$ of prime level $N$, if there is something different from the version worked by Zhou for the Maass forms defined for $\Gamma_0(N)$ on $GL_3$. Notice that the Maass form defined by Zhou (see Page 5 therein) seems not to be a cusp form, while $F=\text{sym}^2 f$ is. Great thanks for spending your time. Much obliged!
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Voronoi formula for the symmetric $L$-function with level $N $
.... So the biggest question is that the exact formula deduced laboriously in the {\bf{Theorem}} above can be directly followed from (2) in Zhou's work??
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Voronoi formula for the symmetric $L$-function with level $N $
@WindomEarle Hi there. Sorry to disturb again. Dear professor, I still have a puzzle needing some helps. As mentioned by Peter Humphries, the classical version for symmetric lift of the newform on $GL_2$ of prime level do not exist, but it seems that the work of Zhou, precisely (2) in the work, [The Voronoi formula on GL(3) with ramification] defined the Maass form for $\Gamma_0(N)$ on $GL_3$. I wander if the form $F=\text{sym}^2 f$ is one Maass form defined by Zhou?
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Voronoi formula for the symmetric $L$-function with level $N $
@WindomEarle Many many thanks. Every thing is clear! Please accept my very warmly warmly thanks for explanation along the time. I have concluded the argument above in my recent draft "Hybrid bounds for $GL_3\times GL_1$ twisted $L$-functions at level aspect" with a deep appreciation. Thank you again for help with no any reward! My name is Fei Hou, it's a good enjoyment to learn the sophisticated insights form you, an excellent expert in number theory! Thanks again and again!!!
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Voronoi formula for the symmetric $L$-function with level $N $
Sorry, I wander if $\phi_{p} $ here is just $\psi_p(x\frac{a}{c})W_{G,p}$, not $\Pi_p$ in the comment next to last above.
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Voronoi formula for the symmetric $L$-function with level $N $
I searched Corbett's paper (arxiv.org/pdf/1807.00716.pdf); the definition for additive character is given in (3). But I didn't know why we have $x\frac{a}{c}$ in $\mathcal{B}_{\Pi_{p},\phi_{p}}$ here. Also I guess \begin{equation} \int_{\mathbb{Q}_p^{\times}} \xi(x)\psi_p(x\frac{a}{c})W_{G,p}\left(\left(\begin{matrix} x&0&0\\0&1&0\\0&0&1\end{matrix}\right)\right)\vert x\vert_p^{-s}d^{\times}x = Z(1-s,\Pi_p)W_{G,p}(1) \nonumber \end{equation} not equaling to $Z(1-s,W_{G,p})$. Besides, it seems that there is no Lemma 2.2 in Corbett's paper. It's weird, the post mentioned Lemma 2.2.
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Voronoi formula for the symmetric $L$-function with level $N $
@WindomEarle Many thanks for improving the post. Obviously, the poster is a top master who would't divulge the name. He/She deserves deeply respect and great gratitude. However, sorry, I still have a question: what $\psi_p(x\frac{a}{c})$ stands for in the formula for $\mathcal{B}_{\Pi_{p},\phi_{p}}$ after the paragraph "The transforms $\mathcal{B}_{\Pi_{p},\phi_{p}}(\cdot)$ are $p$-adic ....". I didn't know its exact definition, and why it equals to one when $(c,N)=1$？I guess maybe it stems form (19) in Corbett's paper. I wander if $\Pi_p$ here is just $\psi_p(x\frac{a}{c})W_{G,p}$?