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Remarks on the symmetric powers of cusp forms on \(\text{GL}(2)\). (English) Zbl 1262.11064

Ginzburg, David (ed.) et al., Automorphic forms and \(L\)-functions I. Global aspects. A workshop in honor of Steve Gelbart on the occasion of his sixtieth birthday, Rehovot and Tel Aviv, Israel, May 15–19, 2006. Providence, RI: American Mathematical Society (AMS); Ramat Gan: Bar-Ilan University (ISBN 978-0-8218-4706-0/pbk). Israel Mathematical Conference Proceedings. Contemp. Math. 488, 237-256 (2009).
In this paper the author studies functorial liftings between automorphic representations of \(\text{GL}_2(\mathbb A_F)\) (\(F\) a number field, \(\mathbb A_F\) its adele ring) and automorphic representations of \(\text{GL}_{m+1}(\mathbb A_F)\), \((m\geq 2)\) attached to the symmetric power representations \(\text{sym}^m: \text{GL}_2(\mathbb C) \to \text{GL}_{m+1}(\mathbb C)\). Specifically, if \(\pi\) is an irreducible automorphic representation of \(\text{GL}_2(\mathbb A_F)\) then, using the local Langlands correspondence at all places, one may construct the lifted representation \(\text{sym}^m(\pi)\) of \(\text{GL}_{m+1}(\mathbb A_F)\) for each \(m\). One may then ask first whether \(\text{sym}^m(\pi)\) is an automorphic representation, and second whether it is a cuspidal automorphic representation.
An isobaric representation \(\pi\) of \(\text{GL}_2(\mathbb A_F)\) is said to be
dihedral if is is automorphically induced by an idele class character of a quadratic extension of \(F\),
tetrahedral if it is not dihedral but its symmetric square lift is automorphically induced by an idele class character of a cubic extension of \(F\),
octahedral if it is not dihedral or tetrahedral but the base change \(BC_{K/F}(\text{sym}^3(\pi))\) of its symmetric cube lift is non-cuspidal for some quadratic extension \(K\) of \(F\), and
solvable polyhedral if any of the above holds.
The main results of this paper may now be stated. For any cuspidal representation \(\pi\) which is not solvable polyhedral:
if \(\text{sym}^5(\pi)\) is modular, then it is cuspidal,
if \(\text{sym}^5(\pi)\) and \(\text{sym}^6(\pi)\) are both modular, then \(\text{sym}^6(\pi)\) is non-cuspidal if and only if \[ \text{sym}^5(\pi) \simeq \text{Ad}(\pi') \boxtimes \pi \otimes \omega^2, \] where \(\omega\) is the central character of \(\pi\) and \(\pi'\) is some other cuspidal automorphic representation of \(\text{GL}_2(\mathbb A_F)\).
for \(m \geq 6,\) \(\text{sym}^m(\pi)\) is cuspidal if and only if \(\text{sym}^6(\pi)\) is cuspidal, provide either
\(\text{sym}^j(\pi)\) is modular for every \(j \leq 2m,\) or
\(\pi \boxtimes \tau\) is modular for any irreducible cuspidal automorphic representation of \(\text{GL}_r(\mathbb A_F)\) with \(r \leq \lfloor \frac m2 + 1\rfloor.\)
Finally, if \(\pi\) is an automorphic representation of \(\text{GL}(2, \mathbb A_\mathbb Q)\) attached to a holomorphic newform of weight \(k \geq 2\) which is not CM, then \(\text{sym}^m(\pi)\) is cuspidal whenever it is modular.
For the entire collection see [Zbl 1166.11002].

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F80 Galois representations
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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