Periods and Reciprocity I

Given $\mathbf{F}$ a number field with ring of integers ${\mathcal{O}}_{\mathbf{F}}$ and $\mathfrak{p},\mathfrak{q}$ two squarefree and coprime ideals of ${\mathcal{O}}_{\mathbf{F}}$, we prove a reciprocity relation for the first moment of the triple product $L$-functions $L(\pi \otimes {\pi }_1\otimes {\pi }_2,\frac{1}{2})$ twisted by ${\lambda }_{\pi }(\mathfrak{p})$, where ${\pi }_1$ and ${\pi }_2$ are a fixed unitary automorphic representation of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2({\mathbb{A}}_{\mathbf{F}})$ with ${\pi }_1$ cuspidal and $\pi$ runs through unitary automorphic representations of conductor dividing $\mathfrak{q}$. The method uses adelic integral representations of $L$-functions and the symmetric identity is established for a particular period. Finally, the integral period is connected to the second moment via Parseval formula.