Rankin–selberg Integrals for Local Symmetric Square Factors on Gl(2)

Abstract

Let π be an irreducible admissible (complex) representation of GL(2) over a non‐Archimedean characteristic zero local field with odd residual characteristic. In this paper, we prove the equality between the local symmetric square L‐function associated to π arising from integral representations and the corresponding Artin L‐function for its Langlands parameter through the local Langlands correspondence. With this in hand, we show the stability of local symmetric γ‐factors attached to π under highly ramified twists.

The local period integrals and essential vectors

“…At this point, we fail to formulate the agreement $$\mathcal {L}(s,\pi , \mathrm{Sym}^2)=L(s,\pi , \mathrm{Sym}^2)$$, which implies the identity $$$$\begin{equation*} L(s,\pi \times \pi )=L(s,\pi , \wedge ^2)L(s,\pi , \mathrm{Sym}^2) \end{equation*}$$$$with $$\mathcal {L}(s,\pi , \mathrm{Sym}^2)$$ appearing in (5.3). Nevertheless, the statement has been recently confirmed for $$m=2$$ [33]. Taking it for granted, it can be seen that $$L(s,\pi ,\mathrm{Sym}^2)$$ is holomorphic at $$s=1$$.…”

“…Combining Proposition 3.1 with Proposition 5.1, the formal symmetric square L ‐factor can be paraphrased as $$$$\begin{equation*} \mathcal {L}(s,\pi _{ur},\mathrm{Sym}^2)= \prod _{1 \le i \le j \le r} (1-\alpha _i(\varpi )\alpha _j(\varpi )q^{-s})^{-1}. \end{equation*}$$$$Remark At this moment, the equality $$\mathcal {L}(s,\pi _{ur},\mathrm{Sym}^2)=L(s,\pi _{ur},\mathrm{Sym}^2)$$ is not known but for G 2 [33]. To get around this, we obtain the “formal symmetric square L ‐factor” $$\mathcal {L}(s,\pi _{ur},\mathrm{Sym}^2)$$, which is enough at least for describing the main result of our interest.…”

“…Remark 6.2. At this moment, the equality (𝑠, 𝜋 𝑢𝑟 , Sym 2 ) = 𝐿(𝑠, 𝜋 𝑢𝑟 , Sym 2 ) is not known but for 𝐺 2 [33]. To get around this, we obtain the "formal symmetric square 𝐿-factor" (𝑠, 𝜋 𝑢𝑟 , Sym 2 ), which is enough at least for describing the main result of our interest.…”

“…However, with the exception of Bump–Ginzburg period cases, we do impose the distinction assumption in nonunitary generic cases. The main rationale to rule out θ‐distinguished representations is that the equality $$L(s,\pi ,\mathrm{Sym}^2)=\mathcal {L}(s,\pi ,\mathrm{Sym}^2)$$ is solely available for $$n=2$$ [33], where $$L(s,\pi ,\mathrm{Sym}^2)$$ is defined by the normalized generator of a fractional ideal as explained in [58, Definition 3.12]. Therefore, we cannot guarantee the holomorphy of Bump–Ginzburg integrals at $$s=1$$ (see Remark 6.9).…”

The local converse theorem for odd special orthogonal and symplectic groups in positive characteristic

“…In the mid 1990's, this partial Bessel function was explored by Baruch [6] to establish the stability of gamma factors and the local converse theorem for U (2,1). A slightly different modification of Howe vectors was pursued by Cogdell and Piatetski-Shapiro, while they treated the stability of gamma factors for SO 2r+1 (F ) [11] defined by identical Rankin-Selberg integrals in Theorem A as a part of their program to establish functorial transfer from generic forms on SO 2r+1 (A k ) to GL 2r (A k ), where A k is the ring of adeles of a number field k. Afterwords, Howe vectors has been implemented in a flurry of work on the stability of numerous local factors and local converse theorems on lower rank groups via the Rankin-Selberg method [27,28,56,57]. All these previous work has consistently involved in the partial Bessel functions for Weyl elements of Bessel distance 1 with the identity (see [14, §5.2.3], [28,Remark 1.2] and [57, §3.2]).…”