Periods and Reciprocity I

Abstract

Abstract Given $\boldsymbol{F}$ a number field with ring of integers ${\mathcal{O}}_{\boldsymbol{F}}$ and ${\mathfrak{p}},{\mathfrak{q}}$ two coprime and squarefree ideals of ${\mathcal{O}}_{\boldsymbol{F}}$, we prove a reciprocity relation for the 1st 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 fixed unitary automorphic representations of $\textrm{PGL}_2({\mathbb{A}}_{\boldsymbol{F}})$ 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 moment via Parseval formula.