Abstract Let G be the F-rational points of the symplectic group Sp2n, where F is a non-Archimedean local field of characteristic 0. Cogdell, Kim, Piatetski-Shapiro, and Shahidi constructed local Lang- lands functorial lifting from irreducible generic representations of G to irreducible representations of GL2n+1(F). Jiang and Soudry constructed the descent map from irreducible supercuspidal repre- sentations of GL2n+1(F) to those of G, showing that the local Langlands functorial lifting from the irreducible supercuspidal generic representations is surjective. In this paper, based on above results, using the same descent method of studying SO2n+1 as Jiang and Soudry, we will show the rest of local Langlands functorial lifting is also surjective, and for any local Langlands parameter , we construct a representation such that and ¾ have the same twisted local factors. As one application, we prove the G-case of a conjecture of Gross-Prasad and Rallis, that is, a local Langlands parameter is generic, i.e., the representation attached to is generic, if and only if the adjoint L-function of is holomorphic at s = 1. As another application, we prove for each Arthur parameter , and the corresponding local Langlands parameter , the representation attached to is generic if and only if is tempered.
Families of abelian surfaces with real multiplication over Hilbert modular surfaces
