Timeline for Is the adjoint L-function on GL(m) holomorphic?
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| Sep 6, 2016 at 15:33 | comment | added | GH from MO | @JosephHundley: Thanks for your comments! | |
| Sep 6, 2016 at 13:25 | comment | added | Joseph Hundley | (3) I should perhaps mention that the integral representation in the case m=5 is not really complete. The unramified computation is reduced to an identity which we are able to check in some special cases but not prove in general. | |
| Sep 6, 2016 at 13:24 | comment | added | Joseph Hundley | (1) If I'm not mistaken, the Flicker result only applies to representations with a supercuspidal component. Which means that-- again, if I'm not mistaken-- even the case m=3 is not completely solved. (2) The integral representations mentioned all involve Eisenstein series which have poles, so they do not give holomorphy automatically. A useful idea for showing that poles of the Eisenstein series are not inherited by the L function is given in Ginzburg-Jiang JNT 82 pp. 256--287. But one still needs to worry about Archimedean and ramified places. | |
| Aug 23, 2016 at 0:27 | comment | added | GH from MO | @QingZ: You are right, I should have checked this reference more carefully. | |
| Aug 22, 2016 at 17:53 | comment | added | Q-Zh | @GHfromMO It seems that in Ginzburg's invent. paper (1991, 571-588, the GL(3) case), he even did not show the local zeta integral is non-vanishing, and thus he even did not show the global L-function has meromorphic continuation. What Ginzburg did is the following: he established a global integral and showed that it is Eulerian, and computed the local integral at unramified places and thus showed the local zeta integral computes the adjoint L-function. I could not find the reference for the local theory and thus I do not know where it is showed that the global adjoint L-function is entire. | |
| Jun 10, 2014 at 15:05 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Jun 10, 2014 at 15:03 | comment | added | GH from MO | @GFS: The results you mention prove the conjecture for $m=3,4,5$ via the integral representation they exhibit. As I told above, the cases $m=3,4$ were also proved by Flicker by a different method. As Bump-Ginzburg remark in their 2008 Crelle paper, the holomorphicity of the adjoint $L$-function would follow from Langlands functoriality. They also mention that their method makes use of an exceptional group, hence it does not generalize to $GL(m)$. So I still think that the general case is widely open. | |
| Jun 10, 2014 at 4:47 | comment | added | GFS | I doubt that. I see Ginzburg's work on adjoint m=3, Bump&Ginzburg on adjoint m=4, Ginzburg&Hundley on adjoint m=5, with some integral representation of L-functions. Does that prove that adjoint L-functions are holomorphic? | |
| Jun 9, 2014 at 7:33 | history | answered | GH from MO | CC BY-SA 3.0 |