Symmetric cube $L$-functions for $\text{GL}_2$ are entire

Henry H. Kim and Freydoon Shahidi

Review from Zentralblatt MATH:

The authors prove the holomorphy of the symmetric cube $L$-functions for non monomial cusp forms. Their method is representation-theoretic and relies on the works of {\it G. Muic} [Duke Math. J. 90, 465-493 (1997; Zbl 0883.11020)].

Since the reviewer is not an expert, I will quote from the authors' introduction:

` To be more precise, let $F$ be a number field whose ring of adeles is $\Bbb A = \Bbb A_F$. Let $\pi = \otimes_v \pi_v$ be a cuspidal (unitary) representation of $GL_2(\Bbb A)$. Let $S$ be a finite set of places of $F$ such that for $V \not\in S$, $\pi_v$ is unramified. For each $v\not \in S$, let $$ t_v = \left\{ \pmatrix \alpha_v & 0
0 & \beta_v\endpmatrix \right\} $$ denote the semisimple conjugacy class of $GL_2(\Bbb C)$ defining $\pi_v$. We recall that if $\pi$ is attached to a classical modular form of weight $k$ on the upper half-plane for which the Fourier coefficient at $p$ is $a_p$, then $$ \aligned a_p &= p^{\frac{k-1}{2}}(\alpha_p + \beta_p)
&= p^{\frac{k-1}{2}}(\alpha_p + \alpha_p^{-1}). \endaligned $$ Fix a positive integer $m$. Denote by $r_m=\text{sym}^m(\rho_2)$, the $m$th symmetric power representation of the standard representation $\rho_2$ of $GL_2(\Bbb C) = \ ^LGL_2^0$, an irreducible representation of dimension $m+1$. Set $$ L_S(s,\pi,r_m) = \prod_{v\not \in S} L(s,\pi_v,r_m), $$ where $$ \aligned L(s,\pi_v,r_m) &=\det(I-r_m(t_v)q_v^{-s})^{-1}
&= \prod_{0 \le j \le m} (1- \alpha_v^j\beta_v^{m-j} q_v^{-s})^{-1} \endaligned $$ is the Langlands $L$-function attached to $\pi_v$ and $r_m$. Recall that if $O_v$ and $P_v$ are the ring of integers of $F_v$ and its unique maximal ideal then $q_v= \text{card}(O_v/P_v)$. A cuspidal representation $\pi$ of $GL_2(\Bbb A)$ is called monomial if $\pi \cong \pi \otimes \eta$ for some nontrivial grössencharacter $\eta$, necessarily quadratic.'

A weakened form of the main result of this paper is: If $\pi$ is not monomial, then the partial symmetric cube $L$-function $L_S(s,\pi,r_3)$ is entire.

Reviewed by Kevin L.James

Keywords: holomorphy; symmetric cube $L$-function; non-monomial cusp forms

Classification (MSC2000): 11F66 11F70

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