Timeline for Jacquet's approach to Rankin--Selberg L-functions
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 12, 2015 at 17:04 | comment | added | Kimball | @DavidLoeffler Oh, do you mean to essentially look at the $SL_2(\mathbb Q)$ translates of a modular form? I guess this won't quite give you all of $\pi_f$, but a dense subset, which should be enough to get you Jacquet's global result, but it's not clear to me if you can translate his local analysis into this setting. | |
| Dec 12, 2015 at 15:08 | comment | added | Kimball | @DavidLoeffler Ah, I didn't realize you could model $\pi_f$ classically. What does this classical model look like? And what does (upper half-plane)*($\widehat{\mathbb Z}^*$) mean? | |
| Dec 12, 2015 at 10:14 | comment | added | David Loeffler | .. and unless I'm very much mistaken, the Petersson product of these Eisenstein series with vectors from (our model of) $\pi_f$ and $\pi'_f$ are exactly Jacquet's adelic integrals, so the GCD over all choices of the two cusp forms and the Schwartz function is (by definition) the $L$-function. | |
| Dec 12, 2015 at 10:13 | comment | added | David Loeffler | ... The point is that test vectors at infinity are explicitly known, and one can write down the finite parts of the automorphic representations $\pi$ and $\pi'$ as explicit spaces of holomorphic functions on the upper half-plane (or more accurately on (upper half-plane) * ($\widehat{\mathbf{Z}}^*)$, to keep track of the component groups). One can also write down a collection of (non-holomorphic) Eisenstein series depending on a choice of a Schwartz function on $\mathbf{A}_f^2$; | |
| Dec 12, 2015 at 10:11 | comment | added | David Loeffler | So you're basically saying that my question doesn't deserve to be answered because it's morally unsound? (Don't worry, I'm only joking! :-) ). But I think that some "middle way" between Jacquet's approach and the adelic approach does exist, and since posting my question I sat down and worked out the details. | |
| Dec 11, 2015 at 16:58 | history | edited | Jeremy Rouse | CC BY-SA 3.0 |
added 4 characters in body
|
| Dec 11, 2015 at 16:34 | history | answered | Kimball | CC BY-SA 3.0 |