Siegel Modular Forms and Representations

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 Γn = Sp(2n, Z), generalizing the well-known procedure for n = 1. This will show that the so-called “standard” and “spin” 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 F4, is then used to establish meromorphic continuation for the spin L-function when n = 3. Introduction Let f be a Siegel modular form of degree n for the full modular group Γn = Sp(2n,Z). If f is an eigenfunction for the action of the Hecke algebra, then there are two L-functions associated with f . Let a0, a1, . . . , an be the Satake parameters of f , and define the standard L-function L1(s, f) = ∏