An explicit Watson-Ichino formula with CM newforms (Q6547201)

In the special case for \(\mathrm{GL}(2)\) and the split cubic etale \(\mathbb{Q}\)-algebra \(\mathbb{Q}^3\), \textit{A. Ichino}'s formula [Duke Math. J. 145, No. 2, 281--307 (2008; Zbl 1222.11065)] for triple products gives a factorization of \( \left| \int_{\mathbb{A}^{\times} \mathrm{GL}(2, \mathbb{Q}) \backslash \mathrm{GL}(2, \mathbb{A})} \varphi_1 \varphi_2 \varphi_3 \,\mathrm{d}g \right|^2 \bigg/ \prod_{i=1}^3 \| \varphi_i \|_{\mathrm{Pet}}^2 \) into local terms multiplied with \( L\left( \frac{1}{2}, \pi_1 \times \pi_2 \times \pi_3 \right) \), together with certain constant and adjoint L-values at \(1\), where each \(\pi_i\) is a unitary cuspidal automorphic representation of \( \mathrm{GL}(2, \mathbb{A}) \) and \( \varphi_i \in \pi_i \). This work is built on earlier works of Watson \textit{et al.} However, the local terms in his formula are not so easy to describe explicitly.\N\NWorking with \( \pi_3 = \overline{\pi_2} \) in the classical language, the triple product L-function \( L(s, f \otimes g \otimes \overline{g}) \) factorizes as \( L(s, f) L(s, f \otimes \mathrm{Ad}(g)) \) where we assume \(f\) is a Hecke-Maass form and \(g\) is a dihedral Maass newform. \textit{P. Humphries} and \textit{R. Khan} established in [Geom. Funct. Anal. 30, No. 1, 34--125 (2020; Zbl 1462.11041)] an exact formula for the central L-value, and applied it to study fourth moments of such forms \(g\).\N\NIn this work, the author follows the aforementioned work and \textit{Y. Hu}, Am. J. Math. 139, No. 1, 215--259 (2017; Zbl 1393.11041)] by proving an explicit version of Ichino's formula for \(L \left(\frac{1}{2}, f \otimes \mathrm{Ad}(g)\right) \), in the case where \( g = g_\Omega \) is a CM newform arising from a quadratic imaginary field \(E\) with negative fundamental discriminant \(D\) and a Hecke character \(\Omega\) of \( E^{\times} \backslash \mathbb{A}_E^{\times} \) such that \(\Omega|_{\mathbb{A}^{\times}} \) is trivial, \(\Omega\) is everywhere unramified and its Archimedean component is of the form \(z \mapsto (z/\bar{z})^n \) for some \(n \in \mathbb{Z}\). There are also conditions imposed on the level and Nebentypus of \(f\). The precise statement is in Theorem 1.1 of this work. This explicit formula is expected to be useful for studying quantum variance of CM newforms.