Periods and Reciprocity II

Let $\mathbf{F}$ be a number field and $\mathfrak{q},\mathfrak{l}$ two coprime integral ideals with $\mathfrak{q}$ squarefree and $\pi_1,\pi_2$ two fixed unitary automorphic representations of $\mathrm{PGL}_2(\mathbb{A}_{\mathbf{F}})$ unramified at all finite places. In this paper, we use regularized integrals to obtain a formula that links the first moment of $L(\pi\otimes\pi_1\otimes\pi_2,\tfrac{1}{2})$ twisted by the Hecke eigenvalues $\lambda_\pi (\mathfrak{l})$, where $\pi$ runs through unitary automorphic representations of $\mathrm{PGL}_2(\mathbb{A}_{\mathbf{F}})$ with conductor dividing $\mathfrak{q}$, with some spectral expansion of periods over representations of conductor dividing $\mathfrak{l}$. In the special case where $\pi_1=\pi_2=\sigma$, this formula becomes a reciprocity relation between moments of $L$-functions. As applications, we obtain a subconvex estimate in the level aspect for the central value of the triple product $L(\pi\otimes\pi_1\otimes\pi_2,\tfrac{1}{2})$ and a simultaneous non-vanishing result for $L(\mathrm{Sym}^2(\sigma)\otimes \pi,\tfrac{1}{2})$ and $L(\pi,\tfrac{1}{2})$.