Abstract. This paper explicitly describes the procedure of associating an automorphic representation of PGSp(2n, A) with a Siegel modular form of degree n for the full modular group n = Sp(2n, Z), generalizing the well-known procedure for n = 1. This will show that the so-called "standard" and "spinor" L-functions associated with such forms are obtained as Langlands L-functions. The theory of Euler products, developed by Langlands, applied to a Levi subgroup of the exceptional group of type F 4 , is then used to establish meromorphic continuation for the spinor L-function when n = 3.

We establish the Langlands functoriality conjecture for the transfer from the generic spectrum of GSp(4) to GL(4) and give a criterion for the cuspidality of its image. We apply this to prove results toward the generalized Ramanujan conjecture for generic representations of GSp(4).

We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group are not self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose L-groups have classical derived groups. The important transfer from GSp 4 to GL 4 follows from our result as a special case.

Abstract. We study the local L-functions for Levi subgroups in split spinor groups defined via the Langlands-Shahidi method and prove a conjecture on their holomorphy in a half plane. These results have been used in the work of Kim and Shahidi on the functorial product for GL 2 × GL 3 .