The Ramanujan Conjecture for Genus two Siegel modular Forms

Abstract

In this chapter we apply the results obtained in Chaps. 1 and 2 and work of Kottwitz in the special case of the symplectic group of similitudes GSp(4). The main result obtained in this chapter is the proof of the Ramanujan conjecture for GSp(4) for cohomological automorphic forms which are not cuspidal representations associated with parabolic subgroups (CAP). We will assume certain statements regarding the trace formula, the proof of which occupies Chaps. 6–10.

For the moment let G be a connected reductive group over Q, whose derived group is simply connected. Assume that the maximal Q-split torus A_G in the center coincides with the maximal R-split torus in the center. Let K ⊆ G(A_fin) be a sufficiently small compact open subgroup (see Sect. 2.2).