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 $L(s,sigmatimes
chi)$ 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.
This paper studies a class of Abelian varieties that are of $mathrm{GL}_2$-type and with isogenous classes defined over a number field $k$ (i.e., $k$-virtual). We treat both cases when their endomorphism algebras are (1) a totally real field $K$ or (
2) a totally indefinite quaternion algebra over a totally real field $K$. Among the isogenous class of such an Abelian variety, we identify one whose Galois conjugates can be described in terms of Atkin-Lehner operators and certain action of the class group of $K$. We deduce that such Abelian varieties are parametrised by finite quotients of certain PEL Shimura varieties. These new families of moduli spaces are further analysed when they are of dimension $2$. We provide explicit numerical bounds for when they are surfaces of general type. In addition, for two particular examples, we calculate precisely the coordinates of inequivalent elliptic points, study intersections of certain Hirzebruch cycles with exceptional divisors. We are able to show that they are both rational surfaces.
This article shows that for unitary dual reductive pairs the first occurrence of theta lift of an irreducible cuspidal automorphic representation is irreducible. It also proves a refined tower property for theta lifts and the involutive property for twisted theta lifts.
Following the idea of [GJS09] for orthogonal groups, we introduce a new family of period integrals for cuspidal automorphic representations $sigma$ of unitary groups and investigate their relation with the occurrence of a simple global Arthur paramet
er $(chi,b)$ in the global Arthur parameter $psi_sigma$ associated to $sigma$, by the endoscopic classification of Arthur ([Art13], [Mok13], [KMSW14]). The argument uses the theory of theta correspondence. This can be viewed as a part of the $(chi,b)$-theory outlined in [Jia14] and can be regarded as a refinement of the theory of theta correspondences and poles of certain $L$-functions, which was outlined in [Ral91].
