Spectral Reciprocity for the first moment of triple product $L$-functions and applications

Let $F$ be a number field with adele ring ${\mathbb{A}}_F$, ${\pi }_1,{\pi }_2$ be two fixed unitary automorphic representations of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2({\mathbb{A}}_F)$ with finite coprime analytic conductor $\mathfrak{u}$ and $\mathfrak{v}$, $\mathfrak{q},\mathfrak{l}$ be two coprime integral ideals with $(\mathfrak{q}\mathfrak{l},\mathfrak{u}\mathfrak{v})=1$. Following [Zac20], we estimate the first moment of $L(\frac{1}{2},\pi \otimes {\pi }_1\otimes {\pi }_2)$ twisted by the Hecke eigenvalues ${\lambda }_{\pi }(\mathfrak{l})$, where $\pi$ runs over unitary automorphic representations of finite conductor dividing $\mathfrak{u}\mathfrak{v}\mathfrak{q}$. By applying the triple product integrals, spectral decomposition and Plancherel formula, we get a reciprocity formula links the twisted first moment of triple product $L$-functions to the spectral expansion of certain triple product periods over automorphic representations of finite conductor dividing $\mathfrak{l}$. As application, we study the subconvexity problem for the triple product $L$-function in the level aspect and give a subconvex bound for $L(\frac{1}{2},\pi \otimes {\pi }_1\otimes {\pi }_2)$ in terms of the norm of $\mathfrak{q}$.