Timeline for Is it possible $L(\frac{1}{2},\phi \times \phi')=0$ for all $\phi'$?
Current License: CC BY-SA 4.0
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| Dec 3, 2021 at 17:14 | comment | added | Monty | @Peter, Oh, I found some papers related with this. Those are Feigon and PD. Nelson's papers. Thank you! | |
| Dec 3, 2021 at 16:41 | history | edited | Monty | CC BY-SA 4.0 |
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| Dec 3, 2021 at 16:36 | comment | added | Monty | @Peter, Thank you for the comment! Is there a reference for $n=m=2$ case? If it is, I would appreciate if you let me know it. And my precise question is to know the existence of any $m$ such that $m \ge n$ satisfying the above property. | |
| Dec 3, 2021 at 15:50 | comment | added | Peter Humphries | I'm guessing that this isn't known but it's expected that $L(1/2,\pi'\times\pi)$ is usually nonzero. A standard analytic approach to this question would be to fix $\pi$ and look at the average of $|L(1/2,\pi'\times\pi)|^2$ over a family of cuspidal automorphic representations $\pi'$ of orthogonal type (e.g. ordered by analytic conductor). If one could show that this is positive, then nonvanishing would follow for infinitely many $\pi'$. However, aside from low rank examples (e.g. $n = m = 2$), this kind of average is hard to calculate. | |
| Dec 3, 2021 at 13:05 | history | edited | Monty | CC BY-SA 4.0 |
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| Dec 3, 2021 at 12:12 | history | edited | Monty | CC BY-SA 4.0 |
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| Dec 3, 2021 at 11:30 | history | asked | Monty | CC BY-SA 4.0 |