Let $\pi$ be an automorphic representation of $GL\_2$ over a number field. What can I say concerning the order of the pole at $1$ of the $L$-function $L(s, \pi)$? Can we say more about $L(s, \mathrm{Sym}^2 \pi)$ or other constructions depending on $\pi$?


In the case where only holomorphic cusp forms appear, there is no pole. However, what about possible Maass forms?



Poles of $L$-functions associated to Maass forms
Let $\pi$ be a *cuspidal* automorphic representation of $\mathrm{GL}\_n(\mathbb{A}\_F)$, where $F$ is a number field. Then $L(s,\pi)$ is entire and has a zero-free region. The latter is essentially Theorem 5.10 of Iwaniec and Kowalski, while the former is Corollary 13.8 of Godement-Jacquet.


If $\pi$ is not cuspidal, then of course this is not necessarily the case: $\zeta\_F(s)$ is the $L$-function of the trivial representation of $\mathrm{GL}\_1(\mathbb{A}\_F)$, and this has a simple pole at $s = 1$. Of course, if $\pi$ is the isobaric sum $\pi\_1 \boxplus \cdots \boxplus \pi\_r$ of automorphic representations, then $L(s,\pi) = L(s,\pi\_1) \cdots L(s,\pi\_r)$, and if each $\pi\_j$ is cuspidal (and in particular not the trivial representation), this is entire.


For $n = 2$ and cuspidal $\pi$, $\operatorname{sym}^2 \pi$ is a cuspidal automorphic representation of $\mathrm{GL}\_3(\mathbb{A}\_F)$ (and consequently $L(s,\operatorname{sym}^2 \pi)$ is entire) iff $\pi$ is not dihedral; note that $\operatorname{sym}^2 \pi \boxplus \omega\_{\pi} = \pi \otimes \pi$, with $\omega\_{\pi}$ the central character of $\pi$.


If $\pi$ is dihedral, then it is the automorphic induction $\pi(\psi)$ from $\mathrm{GL}\_1(\mathbb{A}\_E) = \mathbb{A}\_E^{\times}$ to $\mathrm{GL}\_2(\mathbb{A}\_F)$ of a Hecke character $\psi$ of $\mathbb{A}\_E^{\times}$, where $E/F$ is a quadratic extension to which one can associate a quadratic Hecke character $\omega\_{E/F}$ of $\mathbb{A}\_F^{\times}$ via class field theory. Furthermore, $\pi(\psi)$ is cuspidal iff $\psi$ does not factor through the norm map $N\_{\mathbb{A}\_E / \mathbb{A}\_F} : \mathbb{A}\_E^{\times} \to \mathbb{A}\_F^{\times}$. The central character $\omega\_{\pi(\psi)}$ of $\pi(\psi)$ is $\psi|\_{\mathbb{A}\_F^{\times}} \omega\_{E/F}$ (noting that the restriction of a Hecke character of $\mathbb{A}\_E^{\times}$ to $\mathbb{A}\_F^{\times}$ is a Hecke character of $\mathbb{A}\_F^{\times}$).


So for dihedral $\pi = \pi(\psi)$, we have that $\operatorname{sym}^2 \pi = \pi(\psi^2) \boxplus \psi|\_{\mathbb{A}\_F^{\times}}$, so that $L(s,\operatorname{sym}^2 \pi) = L(s,\psi^2) L(s,\psi|\_{\mathbb{A}\_F^{\times}})$, where $\pi(\psi^2)$ is the automorphic induction of the Hecke character $\psi^2$ of $\mathbb{A}\_E^{\times}$. In particular, if $\psi|\_{\mathbb{A}\_F^{\times}} = 1$ (so that $\omega\_{\pi(\psi)} = \omega\_{E/F}$), then $L(s,\operatorname{sym}^2 \pi) = L(s,\psi^2) \zeta\_F(s)$, which has a simple pole at $s = 1$. On the other hand, if $\psi^2$ is the trivial character, then $L(s,\psi^2) = \zeta\_E(s) = \zeta\_F(s) L(s,\omega\_{E/F})$, which has a simple pole at $s = 1$; note that $\psi^2 = 1$ implies that $\pi(\psi)$ is noncuspidal, and in fact $\pi(\psi) = \omega\_1 \boxplus \omega\_2$ for some real (quadratic or trivial) Hecke characters $\omega\_1,\omega\_2$ of $\mathbb{A}\_F^{\times}$ satisfying $\omega\_1 \omega\_2 = \omega\_{E/F}$ (for $F = \mathbb{Q}$, this means that $\psi$ is a genus character of the quadratic field $E$).


Note also that $\operatorname{ad} \pi \boxplus 1 = \pi \otimes \widetilde{\pi}$ and $\operatorname{ad} \pi = \operatorname{sym}^2 \pi \otimes \omega\_{\pi}^{-1}$. Thus for dihedral $\pi = \pi(\psi)$, $\operatorname{ad} \pi = (\pi(\psi^2) \otimes \psi^{-1}|\_{\mathbb{A}\_F^{\times}}) \boxplus \omega\_{E/F}$.


