Timeline for $\DeclareMathOperator\sym{sym}$Does $L(s, \sym^2 f \times \sym^2 g)$ have a pole at $s=1$?
Current License: CC BY-SA 4.0
11 events
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| Jul 30, 2021 at 18:06 | comment | added | Stopple | @PeterHumphries Why not make this an answer, so the question does not go 'unanswered'? | |
| Jul 29, 2021 at 23:46 | comment | added | hofnumber | @PeterHumphries Many thanks to your comments and reference! | |
| Jul 29, 2021 at 21:17 | comment | added | Peter Humphries | There should be a bound of the form $L(1,\mathrm{sym}^2 f \otimes \mathrm{sym}^2 g) \ll_{\varepsilon} P^{\varepsilon}$ by an application of the main result of Xiannan Li's Ph.D. thesis. | |
| Jul 29, 2021 at 21:16 | comment | added | Peter Humphries | Yes, there is no pole at $s = 1$. Since $f$ has level $1$ and $g$ has level $P$ (assuming that $g$ has trivial nebentypus), their symmetric square lifts are cuspidal; they are also not dual to each other since $\mathrm{sym}^2 f$ has level $1$ whereas $\mathrm{sym}^2 g$ has level $P$. The Rankin-Selberg convolution of two cuspidal automorphic forms can only have a pole at $s = 1$ if they are dual to each other. | |
| Jul 29, 2021 at 19:08 | comment | added | hofnumber | But I really wanna know if somewhere said about $L(1, \text{sym}^2 f \times \text{sym}^2 g)$? If $f\neq g$, the property of $L(1, \text{sym}^2 f \times \text{sym}^2 g)$ has ever been studied before? | |
| Jul 29, 2021 at 19:05 | comment | added | hofnumber | @ F.C. @GH from MO Really sorry if any offense. I will improve the post editing. | |
| Jul 29, 2021 at 18:58 | history | edited | LSpice | CC BY-SA 4.0 |
`\DeclareMathOperator`; grammar
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| Jul 29, 2021 at 18:48 | comment | added | GH from MO | I deleted the opening phrase "guys" as inappropriate. MathOverflow is not a casual chat site, but a scientific forum. | |
| Jul 29, 2021 at 18:47 | history | edited | GH from MO | CC BY-SA 4.0 |
deleted 8 characters in body
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| Jul 29, 2021 at 18:41 | comment | added | F. C. | Hello. Note that some ladies here may possibly not appreciate to be named by "guys". | |
| Jul 29, 2021 at 17:05 | history | asked | hofnumber | CC BY-SA 4.0 |