Periods and $(\chi,b)$-factors of Cuspidal Automorphic Forms of Symplectic Groups

This is a preliminary version. In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations $\pi$ of symplectic groups $\mathrm{Sp}_{2n}(\mathbb{A})$, which is expected to characterize the right-most pole of the $L$-function $L(s,\pi\times\chi)$ for some order-two character $\chi$ of $F^\times\backslash\mathbb{A}^\times$, and hence to detect the occurrence of a simple global Arthur parameter $(\chi,b)$ in the global Arthur parameter $\psi$ attached to $\pi$.