F-virtual Abelian Varieties and Rallis Inner Product Formula

This thesis consists of two topics. First we study F-virtual Abelian varieties of GL2-type where F is a number field. We show the relation between these Abelian varieties and those defined over F. We compare their l-adic representations and study the modularity of F-virtual Abelian varieties of GL2-type. Then we construct their moduli spaces and in the case where the moduli space is a surface we give criteria when it is of general type. We also give two examples of surfaces that are rational and one that is neither rational nor of general type. Second we prove a crucial case of Siegel-Weil formula for orthogonal groups and metaplectic groups. With this we can compute the pairing of theta functions and show in this case that it is related to the central value of Langlands L-function. This new case of Rallis inner product formula enables us to relate nonvanishing of L-value to the nonvanishing of theta lifting.