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Let $\pi$ be an automorphic representation on $\mathrm{GL}(m)/\mathbb{Q}$. Define $$L(s,\pi,\mathrm{Ad}):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an $L$-function with Euler product of degree $m^2-1$.

Is $L(s,\pi,\mathrm{Ad})$ holomorphic on the entire complex plane? As far as I know, this is known for $m=2$ by the work Shimura and Gelbart-Jacquet. What is know on the case of $m>2$?

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I think this is widely open. Flicker has a conditional result under certain cases of the Artin conjecture for Artin $L$-functions, see the Theorem on Page 232 of Pacific J. Math. 154 (1992). In particular, his result shows that the adjoint $L$-function is entire for $m=2,3,4$.

Added. As GFS remarked, the work of Ginzburg et al. shows, by the method of integral representations, that the adjoint $L$-function is entire for $m=3,4,5$.

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  • $\begingroup$ 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? $\endgroup$
    – GFS
    Commented Jun 10, 2014 at 4:47
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    $\begingroup$ @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. $\endgroup$
    – GH from MO
    Commented Jun 10, 2014 at 15:03
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    $\begingroup$ @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. $\endgroup$
    – Q-Zh
    Commented Aug 22, 2016 at 17:53
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    $\begingroup$ (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. $\endgroup$
    – Joseph Hundley
    Commented Sep 6, 2016 at 13:24
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    $\begingroup$ (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. $\endgroup$
    – Joseph Hundley
    Commented Sep 6, 2016 at 13:25

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