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

In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations $sigma$ of symplectic groups $mathrm{Sp}_{2n}(mathbb{A})$, which detects the right-most pole of the $L$-function $L(s,sigmatimeschi)$ for some character $chi$ of $F^timesbackslashmathbb{A}^times$ of order at most $2$, and hence the occurrence of a simple global Arthur parameter $(chi,b)$ in the global Arthur parameter $psi$ attached to $sigma$. We also give a characterisation of first occurrences of theta correspondence by (regularised) period integrals of residues of certain Eisenstein series.