Timeline for Global Waldspurger packet is finite or infinite?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 6, 2022 at 13:18 | vote | accept | Andrew | ||
| Jan 6, 2022 at 13:16 | comment | added | Andrew | oh, thank you very much! The theta lift $\Theta(\pi \otimes \chi_a)$ is with respect to additive character $\psi_a$ which is related to $a$. I confused that all theta lifts are with respect to the same additive character $\psi$. If it is the case, all theta lift should be different. However, since additive characters in the theta liftings also vary as quadratic characters $\chi$‘s vary, the set of all theta lifts should be finite though $\pi \otimes \chi_a$ all all different. Thank you very much for pointing out some important points. | |
| Jan 5, 2022 at 17:08 | comment | added | Kimball | @Andrew Ah, I misread your comment. See revised answer. | |
| Jan 5, 2022 at 17:07 | history | edited | Kimball | CC BY-SA 4.0 |
added 375 characters in body
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| Jan 5, 2022 at 15:30 | comment | added | Andrew | If we choose $\pi$ as good $\pi_D$ such that $\pi_D$ has infinitely many quadratic twists whose central $L$-values are non-zero, then from the above decompostion, $L_{\pi}^2$ has infinitely many irreducible summands because if $L(\frac{1}{2},\pi \times \chi) \ne 0$ for some quadratic character $\chi$, then the theta lifting $\Theta(\pi \otimes \chi)$ is also nonzero and are all distinguished for different $\chi$'s. I am afraid if these two facts I know about the decomposition of $L_{\pi}^2$ is wrong. | |
| Jan 5, 2022 at 15:29 | comment | added | Andrew | Since $\epsilon_v=1$ for almost $v$, then $L_{\pi}^2$, the full near equivalent class associated to $\pi$, should be the finite sum of such automorphic representations of $Mp_2$. On the other hands, it is known that $L_{\pi}^2:=\sum_{a \in F^{\times} \backslash F^{\times^2}} \Theta(\pi \otimes \chi_a)$. | |
| Jan 5, 2022 at 15:18 | comment | added | Andrew | thanks for such kind reply! But due to my stupidness, it is not still clear to me. For the definition of global Waldspurger packet of $\pi$ I mean is the set of irreducible representation of $Mp_2$ associated to $\pi$ though there is a correspondence between irreducible representation of $Mp_2$ and the set of all irreducible cuspidal representation of $D^{\times}$ corresponding to $\pi$ via JL correspondence. It is known that the global $A$-packet of $Mp_2$ associated to $\pi$ consists $\otimes \sigma_v^{\epsilon_v}$ such that $\prod_v \epsilon_v=\epsilon(\frac{1}{2},\pi)$. | |
| Jan 5, 2022 at 14:04 | history | answered | Kimball | CC BY-SA 4.0 |