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Pam Gu : A family of period integrals related to triple product $L$-functions



Let $F$ be a number field with ring of adeles $\mathbb{A}_F$. Let $r_1,r_2,r_3$ be a triple of positive integers and let $\pi:=\otimes_{i=1}^3\pi_i$ where the $\pi_i$ are all cuspidal automorphic representations of $\mathrm{GL}_{r_i}(\mathbb{A}_F)$. We denote by $L(s,\pi, \otimes^3)=L(s, \pi_1\times \pi_2 \times \pi_3)$ the corresponding triple product $L$-function. It is the Langlands $L$-function defined by the tensor product representation $\otimes^3:{}^L(\mathrm{GL}_{r_1} \times \mathrm{GL}_{r_2} \times \mathrm{GL}_{r_3}) \to \mathrm{GL}_{r_1r_2r_3}(\mathbb{C})$. In this talk I will present a family of Eulerian period integrals, which are holomorphic multiples of the triple product -function in a domain that nontrivially intersects the critical strip. We expect that they satisfy a local multiplicity one statement and a local functional equation. This is joint work with Jayce Getz, Chun-Hsien Hsu and Spencer Leslie.

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