Timeline for Factorization of global Waldspurger's integrals and connection to central L-values
Current License: CC BY-SA 4.0
9 events
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| Sep 6 at 5:46 | comment | added | David Loeffler | I wouldn't call (2) a "factorisation" per se, because $L(\pi_E, 1/2)$ is outside the region of convergence of the Euler product formula. So you shouldn't think of (2) purely as a product of local terms: you need the analytic continuation of the L-function to make sense of it. | |
| Sep 6 at 0:09 | comment | added | Peter Humphries | Both $\int_{\mathbb{A}_F^{\times} E^{\times} \backslash \mathbb{A}_E^{\times}} \langle \pi(h) \cdot f_1,f_2\rangle \, dh$ and $\prod_v \int_{F_v^{\times} \backslash E_v^{\times}} \langle \pi_v(h_v) \cdot f_{1,v},f_{2,v}\rangle \, dh_v$ define elements of $\mathrm{Hom}_{\mathbb{A}_E^{\times}}(\pi \otimes \widetilde{\pi},1)$, which is one-dimensional, so they must be equal up to a scalar. | |
| Sep 5 at 22:43 | history | edited | YCor | CC BY-SA 4.0 |
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| Sep 5 at 21:34 | history | edited | Alvin | CC BY-SA 4.0 |
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| Sep 5 at 20:53 | history | edited | Alvin | CC BY-SA 4.0 |
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| Sep 5 at 20:49 | comment | added | Alvin | Thank you for the response! Both the relative trace formula approach by Hervé Jacquet and Chen Nan, as well as the theta correspondence (or Shimizu lifting), provide proofs of Waldspurger’s formula. What I was trying to clarify is whether the equality of the factorized global period integral with the special value of the L-function (as seen in Waldspurger’s formula) can be viewed as equivalent to the formula itself, or is it better understood as a corollary of Waldspurger’s formula? | |
| Sep 5 at 20:21 | comment | added | Peter Humphries | What do you mean by "direct proof"? There is a proof by Herve Jacquet and Chen Nan that uses the relative trace formula, but I wouldn't say that this is any more direct than using the theta correspondence. | |
| S Sep 5 at 20:02 | review | First questions | |||
| Sep 5 at 22:16 | |||||
| S Sep 5 at 20:02 | history | asked | Alvin | CC BY-SA 4.0 |