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Let $\Delta(z)$ be the modular form associated with Ramanujan $\tau$-function.

For any $k=2,3,...$, $Sym^k\Delta$ is conjectured to be an automorphic form on $\mathrm{GL}(k+1)$ and $L(s, Sym^k\Delta)$ is conjectured to be holomorphic and satisfies a functional equation.

Question 1: For which $k$, has $Sym^k\Delta$ been proved to be automorphic form on $\mathrm{GL}(k+1)$?

Question 2: For which $k$, has $L(s,Sym^k\Delta)$ been proved to be homorphic and satisfies a functional equation?

I know certainly that both questions is YES for $k=2,3,4$ (Gelbart-Jacquet, Kim-Shahidi, Shahidi). But I heard there is progress beyond that because $\Delta$ is a modular form rather than arbitrary automorphic representation on $\mathrm{GL}(2)$.

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  • $\begingroup$ Is my answer below sufficient? If so, could you accept it? $\endgroup$ – 2734364041 Nov 12 '20 at 23:00
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It is indeed true that substantially more is known for holomorphic cusp forms than for general automorphic representations, as a consequence of modularity lifting theorems.

The strongest result so far seems to be due to Clozel and Thorne, who have proved automorphy of $Sym^k(f)$, for $f$ a holomorphic modular form of weight $\ge 2$, for all $k \le 8$. This answers your Questions 1 and 2 for these $k$.

I'm not aware of any cases where Question 2 is known without also answering Question 1 (assuming you want holomorphy on all of $\mathbf{C}$, rather than some smaller domain). But if you're willing to settle for $L(s, Sym^k \Delta)$ being meromorphic on $\mathbf{C}$ with the expected functional equation, then this is known for all $k$, as a consequence of the potential automorphy theorems of Barnet-Lamb, Geraghty, Harris and Taylor. This is discussed in Clozel and Thorne's article.

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  • $\begingroup$ Perhaps it is useful to say briefly why 8 is the current limit. E.g., in the Shahidi method, they ran out of the exceptional Lie groups ($E_8$ provided the 4th symmetric power). Here, I am no expert, but I think the limitation is the first sentence of the last paragraph on page 2 of CT: namely that you have a decomposition into essentially orthogonal reps of degrees 4 and 5, which has been considered/classified by Arthur/Mok/(Moeglin-Waldspurger?), and the analogous does not exist for higher powers? $\endgroup$ – just-a-guest Feb 25 '18 at 23:04
  • $\begingroup$ (cont.) There's also the 4th paragraph of pg 3 CT, which seems to suggest that the level-raising argument only works for $l=5,7$, whereas they actually skirt the above Arthur issue (though similar others remain), by saying that "Schur reps" of unitary type can be handled by deformation theory in $GL_9(\bar Z_7)$. Alternatively, in Math Reviews, Wiese suggests that the failure to handle 10 is more "random", in the sense one needs $k-1$ to be prime, noting that the main result is: $C(l-1)$ implies $C(l+1)$ for $l$ prime, where $C$ is the principal Conjecture in the realm, provable for $l=5,7$. $\endgroup$ – just-a-guest Feb 26 '18 at 4:42
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According to this exciting new preprint of Newton and Thorne (https://arxiv.org/abs/1912.11261), the answer to both Questions 1 and 2 is "all $k\geq 1$".

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